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<span>Search</span> </div> </a> <ul id="toc-Search-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functions_of_more_than_one_variable" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Functions_of_more_than_one_variable"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Functions of more than one variable</span> </div> </a> <ul id="toc-Functions_of_more_than_one_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maxima_or_minima_of_a_functional" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Maxima_or_minima_of_a_functional"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Maxima or minima of a functional</span> </div> </a> <ul id="toc-Maxima_or_minima_of_a_functional-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_relation_to_sets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_relation_to_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>In relation to sets</span> </div> </a> <ul id="toc-In_relation_to_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Argument_of_the_maximum" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Argument_of_the_maximum"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Argument of the maximum</span> </div> </a> <ul id="toc-Argument_of_the_maximum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Maximum and minimum</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 52 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-52" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">52 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D9%86%D9%82%D8%A7%D8%B7_%D8%A7%D9%84%D8%AD%D8%AF%D9%8A%D8%A9" title="النقاط الحدية – Arabic" lang="ar" hreflang="ar" data-title="النقاط الحدية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ekstremum" title="Ekstremum – Azerbaijani" lang="az" hreflang="az" data-title="Ekstremum" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ke%CC%8Dk-ta%CC%8Dt" title="Ke̍k-ta̍t – Minnan" lang="nan" hreflang="nan" data-title="Ke̍k-ta̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/E%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D1%83%D0%BC" title="Eкстремум – Bulgarian" lang="bg" hreflang="bg" data-title="Eкстремум" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%A0xims_i_m%C3%ADnims" title="Màxims i mínims – Catalan" lang="ca" hreflang="ca" data-title="Màxims i mínims" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D1%83%D0%BC" title="Экстремум – Chuvash" lang="cv" hreflang="cv" data-title="Экстремум" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Extr%C3%A9m_funkce" title="Extrém funkce – Czech" lang="cs" hreflang="cs" data-title="Extrém funkce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Zvinyanye" title="Zvinyanye – Shona" lang="sn" hreflang="sn" data-title="Zvinyanye" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Uchafbwyntiau_ac_isafbwyntiau" title="Uchafbwyntiau ac isafbwyntiau – Welsh" lang="cy" hreflang="cy" data-title="Uchafbwyntiau ac isafbwyntiau" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Maksimum_og_minimum" title="Maksimum og minimum – Danish" lang="da" hreflang="da" data-title="Maksimum og minimum" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Extremwert" title="Extremwert – German" lang="de" hreflang="de" data-title="Extremwert" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ekstreemum" title="Ekstreemum – Estonian" lang="et" hreflang="et" data-title="Ekstreemum" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Extremos_de_una_funci%C3%B3n" title="Extremos de una función – Spanish" lang="es" hreflang="es" data-title="Extremos de una función" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Maksimumo_kaj_minimumo" title="Maksimumo kaj minimumo – Esperanto" lang="eo" hreflang="eo" data-title="Maksimumo kaj minimumo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_baten_muturrak" title="Funtzio baten muturrak – Basque" lang="eu" hreflang="eu" data-title="Funtzio baten muturrak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%DB%8C%D8%B4%DB%8C%D9%86%D9%87_%D9%88_%DA%A9%D9%85%DB%8C%D9%86%D9%87" title="بیشینه و کمینه – Persian" lang="fa" hreflang="fa" data-title="بیشینه و کمینه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Extremum" title="Extremum – French" lang="fr" hreflang="fr" data-title="Extremum" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Extremos_dunha_funci%C3%B3n" title="Extremos dunha función – Galician" lang="gl" hreflang="gl" data-title="Extremos dunha función" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B9%EA%B0%92" title="극값 – Korean" lang="ko" hreflang="ko" data-title="극값" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1%D5%B5%D5%AB_%D5%A7%D6%84%D5%BD%D5%BF%D6%80%D5%A5%D5%B4%D5%B8%D6%82%D5%B4" title="Ֆունկցիայի էքստրեմում – Armenian" lang="hy" hreflang="hy" data-title="Ֆունկցիայի էքստրեմում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%89%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%BF%E0%A4%B7%E0%A5%8D%E0%A4%A0_%E0%A4%94%E0%A4%B0_%E0%A4%A8%E0%A4%BF%E0%A4%AE%E0%A5%8D%E0%A4%A8%E0%A4%BF%E0%A4%B7%E0%A5%8D%E0%A4%A0" title="उच्चिष्ठ और निम्निष्ठ – Hindi" lang="hi" hreflang="hi" data-title="उच्चिष्ठ और निम्निष्ठ" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Maksimum_dan_minimum" title="Maksimum dan minimum – Indonesian" lang="id" hreflang="id" data-title="Maksimum dan minimum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%9Atgildi" title="Útgildi – Icelandic" lang="is" hreflang="is" data-title="Útgildi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Massimo_e_minimo_di_una_funzione" title="Massimo e minimo di una funzione – Italian" lang="it" hreflang="it" data-title="Massimo e minimo di una funzione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%A7%D7%99%D7%A6%D7%95%D7%9F" title="נקודת קיצון – Hebrew" lang="he" hreflang="he" data-title="נקודת קיצון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%B0%E0%B2%BF%E0%B2%B7%E0%B3%8D%E0%B2%A0_%E0%B2%AE%E0%B2%A4%E0%B3%8D%E0%B2%A4%E0%B3%81_%E0%B2%95%E0%B2%A8%E0%B2%BF%E0%B2%B7%E0%B3%8D%E0%B2%A0_%E0%B2%AE%E0%B3%8C%E0%B2%B2%E0%B3%8D%E0%B2%AF%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಗರಿಷ್ಠ ಮತ್ತು ಕನಿಷ್ಠ ಮೌಲ್ಯಗಳು – Kannada" lang="kn" hreflang="kn" data-title="ಗರಿಷ್ಠ ಮತ್ತು ಕನಿಷ್ಠ ಮೌಲ್ಯಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D1%83%D0%BC" title="Экстремум – Kazakh" lang="kk" hreflang="kk" data-title="Экстремум" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Maksimums_un_minimums" title="Maksimums un minimums – Latvian" lang="lv" hreflang="lv" data-title="Maksimums un minimums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Massim_e_minim_de_%27na_fonzion" title="Massim e minim de 'na fonzion – Lombard" lang="lmo" hreflang="lmo" data-title="Massim e minim de 'na fonzion" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sz%C3%A9ls%C5%91%C3%A9rt%C3%A9k" title="Szélsőérték – Hungarian" lang="hu" hreflang="hu" data-title="Szélsőérték" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D0%BD%D0%B8_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Екстремни вредности – Macedonian" lang="mk" hreflang="mk" data-title="Екстремни вредности" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Extreme_waarde" title="Extreme waarde – Dutch" lang="nl" hreflang="nl" data-title="Extreme waarde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A5%B5%E5%80%A4" title="極値 – Japanese" lang="ja" hreflang="ja" data-title="極値" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Maksimum_og_minimum" title="Maksimum og minimum – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Maksimum og minimum" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Maksimum_og_minimum" title="Maksimum og minimum – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Maksimum og minimum" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ekstremum" title="Ekstremum – Uzbek" lang="uz" hreflang="uz" data-title="Ekstremum" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ekstremum_funkcji" title="Ekstremum funkcji – Polish" lang="pl" hreflang="pl" data-title="Ekstremum funkcji" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Pontos_extremos_de_uma_fun%C3%A7%C3%A3o" title="Pontos extremos de uma função – Portuguese" lang="pt" hreflang="pt" data-title="Pontos extremos de uma função" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Maxim_%C8%99i_minim" title="Maxim și minim – Romanian" lang="ro" hreflang="ro" data-title="Maxim și minim" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D1%83%D0%BC" title="Экстремум – Russian" lang="ru" hreflang="ru" data-title="Экстремум" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Maximum_and_minimum" title="Maximum and minimum – Simple English" lang="en-simple" hreflang="en-simple" data-title="Maximum and minimum" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Extr%C3%A9m_(funkcia)" title="Extrém (funkcia) – Slovak" lang="sk" hreflang="sk" data-title="Extrém (funkcia)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ekstrem_funkcije" title="Ekstrem funkcije – Slovenian" lang="sl" hreflang="sl" data-title="Ekstrem funkcije" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/%C3%84%C3%A4riarvo" title="Ääriarvo – Finnish" lang="fi" hreflang="fi" data-title="Ääriarvo" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li 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data-title="பெருமம் மற்றும் சிறுமம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D1%83%D0%BC" title="Экстремум – Tatar" lang="tt" hreflang="tt" data-title="Экстремум" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D1%83%D0%BC" title="Екстремум – Ukrainian" lang="uk" hreflang="uk" data-title="Екстремум" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Largest and smallest value taken by a function at a given point</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Extreme value" redirects here. For other uses, see <a href="/wiki/Extreme_value_(disambiguation)" class="mw-disambig" title="Extreme value (disambiguation)">Extreme value (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Maximum" and "Minimum" redirect here. For other uses, see <a href="/wiki/Maximum_(disambiguation)" class="mw-disambig" title="Maximum (disambiguation)">Maximum (disambiguation)</a> and <a href="/wiki/Minimum_(disambiguation)" class="mw-disambig" title="Minimum (disambiguation)">Minimum (disambiguation)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Extrema_example_original.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Extrema_example_original.svg/220px-Extrema_example_original.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Extrema_example_original.svg/330px-Extrema_example_original.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Extrema_example_original.svg/440px-Extrema_example_original.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>Local and global maxima and minima for cos(3π<i>x</i>)/<i>x</i>, 0.1≤<i> x </i>≤1.1</figcaption></figure> <p>In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the <b>maximum</b> and <b>minimum</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> are, respectively, the greatest and least value taken by the function. Known generically as <b>extremum</b>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> they may be defined either within a given <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">range</a> (the <i>local</i> or <i>relative</i> extrema) or on the entire <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> (the <i>global</i> or <i>absolute</i> extrema) of a function.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> was one of the first mathematicians to propose a general technique, <a href="/wiki/Adequality" title="Adequality">adequality</a>, for finding the maxima and minima of functions. </p><p>As defined in <a href="/wiki/Set_theory" title="Set theory">set theory</a>, the maximum and minimum of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> are the <a href="/wiki/Greatest_and_least_elements" class="mw-redirect" title="Greatest and least elements">greatest and least elements</a> in the set, respectively. Unbounded <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a>, such as the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>, have no minimum or maximum. </p><p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, the corresponding concept is the <a href="/wiki/Sample_maximum_and_minimum" title="Sample maximum and minimum">sample maximum and minimum</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A real-valued <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>f</i> defined on a <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> <i>X</i> has a <b>global</b> (or <b>absolute</b>) <b>maximum point</b><span class="anchor" id="Global_maximum_point"></span><span class="anchor" id="Absolute_maximum_point"></span><span class="anchor" id="Maximum_point"></span> at <i>x</i><sup>∗</sup>, if <span class="nowrap"><i>f</i>(<i>x</i><sup>∗</sup>) ≥ <i>f</i>(<i>x</i>)</span> for all <i>x</i> in <i>X</i>. Similarly, the function has a <b>global</b> (or <b>absolute</b>) <b>minimum point</b><span class="anchor" id="Global_minimum_point"></span><span class="anchor" id="Absolute_minimum_point"></span><span class="anchor" id="Minimum_point"></span> at <i>x</i><sup>∗</sup>, if <span class="nowrap"><i>f</i>(<i>x</i><sup>∗</sup>) ≤ <i>f</i>(<i>x</i>)</span> for all <i>x</i> in <i>X</i>. The value of the function at a maximum point is called the <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="maximum_value"></span><span class="vanchor-text">maximum value</span></span></b> of the function, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1e9b1277019f9867f6c9cb856485f3f430ba0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.553ex; height:2.843ex;" alt="{\displaystyle \max(f(x))}"></span>, and the value of the function at a minimum point is called the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="minimum_value"></span><span class="vanchor-text">minimum value</span></span></b> of the function, (denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min(f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04384f84f55ba4e5a9d87b3948d30c9ed06200ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.102ex; height:2.843ex;" alt="{\displaystyle \min(f(x))}"></span> for clarity). Symbolically, this can be written as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b79e955b57dd7aada93b8afd459996ae941d480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}"></span> is a global maximum point of function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72b93833f90a2d703ef1c01556e957bb187f351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.135ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} ,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2272de3591ea6428f4e6d0620cffb5e63cba762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.274ex; height:2.843ex;" alt="{\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).}"></span></dd></dl> <p>The definition of global minimum point also proceeds similarly. </p><p>If the domain <i>X</i> is a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, then <i>f</i> is said to have a <b>local</b> (or <b>relative</b>) <b>maximum point</b><span class="anchor" id="Local_maximum_point"></span><span class="anchor" id="Relative_maximum_point"></span> at the point <i>x</i><sup>∗</sup>, if there exists some <i>ε</i> > 0 such that <span class="nowrap"><i>f</i>(<i>x</i><sup>∗</sup>) ≥ <i>f</i>(<i>x</i>)</span> for all <i>x</i> in <i>X</i> within distance <i>ε</i> of <i>x</i><sup>∗</sup>. Similarly, the function has a <b>local minimum point</b><span class="anchor" id="Local_minimum_point"></span><span class="anchor" id="Relative_minimum_point"></span> at <i>x</i><sup>∗</sup>, if <i>f</i>(<i>x</i><sup>∗</sup>) ≤ <i>f</i>(<i>x</i>) for all <i>x</i> in <i>X</i> within distance <i>ε</i> of <i>x</i><sup>∗</sup>. A similar definition can be used when <i>X</i> is a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: </p> <dl><dd>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,d_{X})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d_{X})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b63a9cf5929b2fe7fca007865e043bb9004c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.664ex; height:2.843ex;" alt="{\displaystyle (X,d_{X})}"></span> be a metric space and function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669fa4832da4b0b229d77eadb270e95188f2eb10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} }"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b79e955b57dd7aada93b8afd459996ae941d480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}"></span> is a local maximum point of function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists \varepsilon >0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\exists \varepsilon >0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f09b0440cc9530512b4972ab5f3aa697e3616c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.446ex; height:2.843ex;" alt="{\displaystyle (\exists \varepsilon >0)}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6ea2c6488ad08285bb2a20c508de7f2f32a94d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.24ex; height:2.843ex;" alt="{\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).}"></span></dd></dl> <p>The definition of local minimum point can also proceed similarly. </p><p>In both the global and local cases, the concept of a <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="strict_extremum"></span><span class="vanchor-text">strict extremum</span></span></b> can be defined. For example, <i>x</i><sup>∗</sup> is a <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="strict_global_maximum_point"></span><span class="vanchor-text">strict global maximum point</span></span></b> if for all <i>x</i> in <i>X</i> with <span class="nowrap"><i>x</i> ≠ <i>x</i><sup>∗</sup></span>, we have <span class="nowrap"><i>f</i>(<i>x</i><sup>∗</sup>) > <i>f</i>(<i>x</i>)</span>, and <i>x</i><sup>∗</sup> is a <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="strict_local_maximum_point"></span><span class="vanchor-text">strict local maximum point</span></span></b> if there exists some <span class="nowrap"><i>ε</i> > 0</span> such that, for all <i>x</i> in <i>X</i> within distance <i>ε</i> of <i>x</i><sup>∗</sup> with <span class="nowrap"><i>x</i> ≠ <i>x</i><sup>∗</sup></span>, we have <span class="nowrap"><i>f</i>(<i>x</i><sup>∗</sup>) > <i>f</i>(<i>x</i>)</span>. Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. </p><p>A <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> real-valued function with a <a href="/wiki/Compact_space" title="Compact space">compact</a> domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a> (see the graph above). </p> <div class="mw-heading mw-heading2"><h2 id="Search">Search</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=2" title="Edit section: Search"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Finding global maxima and minima is the goal of <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>. If a function is continuous on a closed interval, then by the <a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a>, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one.Minima </p><p>For <a href="/wiki/Differentiable_functions" class="mw-redirect" title="Differentiable functions">differentiable functions</a>, <a href="/wiki/Fermat%27s_theorem_(stationary_points)" title="Fermat's theorem (stationary points)">Fermat's theorem</a> states that local extrema in the interior of a domain must occur at <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> (or points where the derivative equals zero).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the <a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">first derivative test</a>, <a href="/wiki/Derivative_test#Second-derivative_test_(single_variable)" title="Derivative test">second derivative test</a>, or <a href="/wiki/Higher-order_derivative_test" class="mw-redirect" title="Higher-order derivative test">higher-order derivative test</a>, given sufficient differentiability.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>For any function that is defined <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a>, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least). </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Xth_root_of_x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Xth_root_of_x.svg/220px-Xth_root_of_x.svg.png" decoding="async" width="220" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Xth_root_of_x.svg/330px-Xth_root_of_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Xth_root_of_x.svg/440px-Xth_root_of_x.svg.png 2x" data-file-width="512" data-file-height="465" /></a><figcaption>The global maximum of <span class="texhtml"><span class="nowrap"><sup style="margin-right: -0.5em; vertical-align: 0.8em;"><i>x</i></sup>√<span style="border-top:1px solid; padding:0 0.1em;"><i>x</i></span></span></span> occurs at <span class="texhtml"><i>x</i> = <i><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></i></span>.</figcaption></figure> <table class="wikitable"> <tbody><tr> <th>Function</th> <th>Maxima and minima </th></tr> <tr> <td><i>x</i><sup>2</sup></td> <td>Unique global minimum at <i>x</i> = 0. </td></tr> <tr> <td><i>x</i><sup>3</sup></td> <td>No global minima or maxima. Although the first derivative (3<i>x</i><sup>2</sup>) is 0 at <i>x</i> = 0, this is an <a href="/wiki/Inflection_point" title="Inflection point">inflection point</a>. (2nd derivative is 0 at that point.) </td></tr> <tr> <td><big><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{x}]{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{x}]{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b4a0a76158849854a302fc639dfc882ec16008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{x}]{x}}}"></span></big></td> <td>Unique global maximum at <i>x</i> = <i><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></i>. (See figure at right) </td></tr> <tr> <td><i>x</i><sup>−<i>x</i></sup></td> <td>Unique global maximum over the positive real numbers at <i>x</i> = 1/<i>e</i>. </td></tr> <tr> <td><i>x</i><sup>3</sup>/3 − <i>x</i></td> <td>First derivative <i>x</i><sup>2</sup> − 1 and <a href="/wiki/Second_derivative" title="Second derivative">second derivative</a> 2<i>x</i>. Setting the first derivative to 0 and solving for <i>x</i> gives <a href="/wiki/Stationary_point" title="Stationary point">stationary points</a> at −1 and +1. From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum. This function has no global maximum or minimum. </td></tr> <tr> <td> |<i>x</i>|</td> <td>Global minimum at <i>x</i> = 0 that cannot be found by taking derivatives, because the derivative does not exist at <i>x</i> = 0. </td></tr> <tr> <td>cos(<i>x</i>)</td> <td>Infinitely many global maxima at 0, ±2<span class="texhtml mvar" style="font-style:italic;">π</span>, ±4<span class="texhtml mvar" style="font-style:italic;">π</span>, ..., and infinitely many global minima at ±<span class="texhtml mvar" style="font-style:italic;">π</span>, ±3<span class="texhtml mvar" style="font-style:italic;">π</span>, ±5<span class="texhtml mvar" style="font-style:italic;">π</span>, .... </td></tr> <tr> <td>2 cos(<i>x</i>) − <i>x</i></td> <td>Infinitely many local maxima and minima, but no global maximum or minimum. </td></tr> <tr> <td>cos(3<span class="texhtml mvar" style="font-style:italic;">π</span><i>x</i>)/<i>x</i> with <span class="nowrap">0.1 ≤ <i>x</i> ≤ 1.1</span></td> <td>Global maximum at <i>x</i> = 0.1 (a boundary), a global minimum near <i>x</i> = 0.3, a local maximum near <i>x</i> = 0.6, and a local minimum near <i>x</i> = 1.0. (See figure at top of page.) </td></tr> <tr> <td><i>x</i><sup>3</sup> + 3<i>x</i><sup>2</sup> − 2<i>x</i> + 1 defined over the closed interval (segment) [−4,2]</td> <td>Local maximum at <i>x</i> = −1−<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">15</span></span>/3, local minimum at <i>x</i> = −1+<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">15</span></span>/3, global maximum at <i>x</i> = 2 and global minimum at <i>x</i> = −4. </td></tr></tbody></table> <p>For a practical example,<sup id="cite_ref-minimization_maximization_refresher_8-0" class="reference"><a href="#cite_note-minimization_maximization_refresher-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> assume a situation where someone has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 200}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>200</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 200}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198350442fbcb39391818f80732eb8701439bad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 200}"></span> feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is the length, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> is the width, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span> is the area: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x+2y=200}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mo>=</mo> <mn>200</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+2y=200}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/441dd355ed8fa0eb140ee74f053bc221f6d9048c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.236ex; height:2.509ex;" alt="{\displaystyle 2x+2y=200}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2y=200-2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>y</mi> <mo>=</mo> <mn>200</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2y=200-2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74952699b03f5961e3303d3b23088316032cb7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.236ex; height:2.509ex;" alt="{\displaystyle 2y=200-2x}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>200</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7429d8a609e5bdbe2296931996ae5a0c022d7a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.909ex; height:5.343ex;" alt="{\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=100-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>100</mn> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=100-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e420c14cad943cd21a14d73b2ee682f2bfb69883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.911ex; height:2.509ex;" alt="{\displaystyle y=100-x}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=x(100-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>100</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=x(100-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54468a2c8da1263d601178d3c34da9808e0dfba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.38ex; height:2.843ex;" alt="{\displaystyle xy=x(100-x)}"></span></dd></dl> <p>The derivative with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>x</mi> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>100</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>100</mn> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>100</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927ccfa5c8f0fb40bee42957fd803189e72e8466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:25.657ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}}"></span></dd></dl> <p>Setting this equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=100-2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mn>100</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=100-2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c146b77770d725bdd0923a90792f5310abe944e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.081ex; height:2.343ex;" alt="{\displaystyle 0=100-2x}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x=100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x=100}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928347e70fe9590f6fd49a9c23694b79f622a4b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.078ex; height:2.176ex;" alt="{\displaystyle 2x=100}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=50}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>50</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=50}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c97c19892373defe73b8a95b63f3840d574da16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle x=50}"></span></dd></dl> <p>reveals that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=50}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>50</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=50}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c97c19892373defe73b8a95b63f3840d574da16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle x=50}"></span> is our only <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a>. Now retrieve the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">endpoints</a> by determining the interval to which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is restricted. Since width is positive, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span>, and since <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=100-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>100</mn> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=100-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359b5165e59b56b580004ee749c52ea07f68befd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.911ex; height:2.509ex;" alt="{\displaystyle x=100-y}"></span>,</span> that implies that <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<100}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c975b633e15b0e7f27986d41e17014bfcc4fb433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.916ex; height:2.176ex;" alt="{\displaystyle x<100}"></span>.</span> Plug in critical point <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 50}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>50</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 50}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e5f8966bed37734cd86d4fd3c302913bb6d48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 50}"></span>,</span> as well as endpoints <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0572cd017c6d7936a12737c9d614a2f801f94a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}"></span>,</span> into <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=x(100-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>100</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=x(100-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54468a2c8da1263d601178d3c34da9808e0dfba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.38ex; height:2.843ex;" alt="{\displaystyle xy=x(100-x)}"></span>,</span> and the results are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2500,0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2500</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2500,0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e19ae76eed1f2872da577bd5f24e666f871419e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.493ex; height:2.509ex;" alt="{\displaystyle 2500,0,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> respectively. </p><p>Therefore, the greatest area attainable with a rectangle of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 200}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>200</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 200}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198350442fbcb39391818f80732eb8701439bad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 200}"></span> feet of fencing is <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 50\times 50=2500}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>50</mn> <mo>×</mo> <mn>50</mn> <mo>=</mo> <mn>2500</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 50\times 50=2500}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab081230ede8044155c48dec8c781f04209bcbd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.239ex; height:2.176ex;" alt="{\displaystyle 50\times 50=2500}"></span>.</span><sup id="cite_ref-minimization_maximization_refresher_8-1" class="reference"><a href="#cite_note-minimization_maximization_refresher-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Functions_of_more_than_one_variable">Functions of more than one variable</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=4" title="Edit section: Functions of more than one variable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Second_partial_derivative_test" title="Second partial derivative test">Second partial derivative test</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX,_1-.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX%2C_1-.jpg/220px-Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX%2C_1-.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX%2C_1-.jpg/330px-Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX%2C_1-.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX%2C_1-.jpg/440px-Modell_einer_Peanoschen_Fl%C3%A4che_-Schilling_XLIX%2C_1-.jpg 2x" data-file-width="600" data-file-height="450" /></a><figcaption><a href="/wiki/Peano_surface" title="Peano surface">Peano surface</a>, a counterexample to some criteria of local maxima of the 19th century</figcaption></figure><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:MaximumParaboloid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/MaximumParaboloid.png/220px-MaximumParaboloid.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/MaximumParaboloid.png/330px-MaximumParaboloid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/MaximumParaboloid.png/440px-MaximumParaboloid.png 2x" data-file-width="756" data-file-height="567" /></a><figcaption>The global maximum is the point at the top</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:MaximumCounterexample.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/MaximumCounterexample.png/220px-MaximumCounterexample.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/MaximumCounterexample.png/330px-MaximumCounterexample.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/MaximumCounterexample.png/440px-MaximumCounterexample.png 2x" data-file-width="756" data-file-height="567" /></a><figcaption>Counterexample: The red dot shows a local minimum that is not a global minimum</figcaption></figure> <p>For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a <i>local</i> maximum are similar to those of a function with only one variable. The first <a href="/wiki/Partial_derivatives" class="mw-redirect" title="Partial derivatives">partial derivatives</a> as to <i>z</i> (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a <a href="/wiki/Saddle_point" title="Saddle point">saddle point</a>. For use of these conditions to solve for a maximum, the function <i>z</i> must also be <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> throughout. The <a href="/wiki/Second_partial_derivative_test" title="Second partial derivative test">second partial derivative test</a> can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function <i>f</i> defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the <a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a> and <a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a> to prove this by <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">contradiction</a>). In two and more dimensions, this argument fails. This is illustrated by the function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b4a21ac7525541b799f233d44a9501ba76bd72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.704ex; height:3.176ex;" alt="{\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,}"></span></dd></dl> <p>whose only critical point is at (0,0), which is a local minimum with <i>f</i>(0,0) = 0. However, it cannot be a global one, because <i>f</i>(2,3) = −5. </p> <div class="mw-heading mw-heading2"><h2 id="Maxima_or_minima_of_a_functional">Maxima or minima of a functional</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=5" title="Edit section: Maxima or minima of a functional"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a>), then the extremum is found using the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_relation_to_sets">In relation to sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=6" title="Edit section: In relation to sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Maxima and minima can also be defined for sets. In general, if an <a href="/wiki/Ordered_set" class="mw-redirect" title="Ordered set">ordered set</a> <i>S</i> has a <a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a> <i>m</i>, then <i>m</i> is a <a href="/wiki/Maximal_element" class="mw-redirect" title="Maximal element">maximal element</a> of the set, also denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08446be939ab20ba0b116419017fd087c60af3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.634ex; height:2.843ex;" alt="{\displaystyle \max(S)}"></span>. Furthermore, if <i>S</i> is a subset of an ordered set <i>T</i> and <i>m</i> is the greatest element of <i>S</i> with (respect to order induced by <i>T</i>), then <i>m</i> is a <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">least upper bound</a> of <i>S</i> in <i>T</i>. Similar results hold for <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a>, <a href="/wiki/Minimal_element" class="mw-redirect" title="Minimal element">minimal element</a> and <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">greatest lower bound</a>. The maximum and minimum function for sets are used in <a href="/wiki/Database" title="Database">databases</a>, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-<a href="/wiki/Decomposable_aggregation_function" class="mw-redirect" title="Decomposable aggregation function">decomposable aggregation functions</a>. </p><p>In the case of a general <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>, the <b>least element</b> (i.e., one that is less than all others) should not be confused with a <b>minimal element</b> (nothing is lesser). Likewise, a <b><a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a></b> of a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> (poset) is an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> of the set which is contained within the set, whereas a <b>maximal element</b> <i>m</i> of a poset <i>A</i> is an element of <i>A</i> such that if <i>m</i> ≤ <i>b</i> (for any <i>b</i> in <i>A</i>), then <i>m</i> = <i>b</i>. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable. </p><p>In a <a href="/wiki/Total_order" title="Total order">totally ordered</a> set, or <i>chain</i>, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms <i><b>minimum</b></i> and <i><b>maximum</b></i>. </p><p>If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have a maximum or a minimum. For example, the set of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> has no maximum, though it has a minimum. If an infinite chain <i>S</i> is bounded, then the <a href="/wiki/Topological_closure" class="mw-redirect" title="Topological closure">closure</a> <i>Cl</i>(<i>S</i>) of the set occasionally has a minimum and a maximum, in which case they are called the <b>greatest lower bound</b> and the <b>least upper bound</b> of the set <i>S</i>, respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Argument_of_the_maximum">Argument of the maximum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=7" title="Edit section: Argument of the maximum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Arg_max" title="Arg max">Arg max</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Arg_max&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Si_sinc.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Si_sinc.svg/350px-Si_sinc.svg.png" decoding="async" width="350" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Si_sinc.svg/525px-Si_sinc.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Si_sinc.svg/700px-Si_sinc.svg.png 2x" data-file-width="670" data-file-height="533" /></a><figcaption>As an example, both unnormalised and normalised <a href="/wiki/Sinc" class="mw-redirect" title="Sinc">sinc</a> functions above have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {argmax} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>argmax</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {argmax} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a65af95b2160c115b5fb79e16ecc0607fe4774" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.562ex; height:2.009ex;" alt="{\displaystyle \operatorname {argmax} }"></span> of {0} because both attain their global maximum value of 1 at <i>x</i> = 0.<br /><br />The unnormalised sinc function (red) has <i>arg min</i> of {−4.49, 4.49}, approximately, because it has 2 global minimum values of approximately −0.217 at <i>x</i> = ±4.49. However, the normalised sinc function (blue) has <i>arg min</i> of {−1.43, 1.43}, approximately, because their global minima occur at <i>x</i> = ±1.43, even though the minimum value is the same.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the arguments of the maxima (abbreviated <a href="/wiki/Arg_max" title="Arg max">arg max</a> or argmax) and arguments of the minima (abbreviated arg min or argmin) are the input points at which a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> output value is <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maximized and minimized</a>, respectively.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> While the <a href="/wiki/Argument_of_a_function" title="Argument of a function">arguments</a> are defined over the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain of a function</a>, the output is part of its <a href="/wiki/Codomain" title="Codomain">codomain</a>.</div></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Derivative_test" title="Derivative test">Derivative test</a></li> <li><a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a></li> <li><a href="/wiki/Limit_superior_and_limit_inferior" class="mw-redirect" title="Limit superior and limit inferior">Limit superior and limit inferior</a></li> <li><a href="/wiki/Maximum-minimums_identity" title="Maximum-minimums identity">Maximum-minimums identity</a></li> <li><a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">Mechanical equilibrium</a></li> <li><a href="/wiki/Mex_(mathematics)" title="Mex (mathematics)">Mex (mathematics)</a></li> <li><a href="/wiki/Saddle_point" title="Saddle point">Saddle point</a></li> <li><a href="/wiki/Sample_maximum_and_minimum" title="Sample maximum and minimum">Sample maximum and minimum</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="/wiki/Plural" title="Plural"><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">PL</span></span></a>: <b>maxima</b> and <b>minima</b> (or <b>maximums</b> and <b>minimums</b>).</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">PL</span></span>: <b>extrema</b>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFStewart2008" class="citation book cs1"><a href="/wiki/James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew_1"><i>Calculus: Early Transcendentals</i></a></span> (6th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-01166-8" title="Special:BookSources/978-0-495-01166-8"><bdi>978-0-495-01166-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2008&rft.isbn=978-0-495-01166-8&rft.aulast=Stewart&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusearlytra00stew_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaximum+and+minimum" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonEdwards2009" class="citation book cs1"><a href="/wiki/Ron_Larson_(mathematician)" class="mw-redirect" title="Ron Larson (mathematician)">Larson, Ron</a>; Edwards, Bruce H. (2009). <i>Calculus</i> (9th ed.). <a href="/wiki/Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-16702-2" title="Special:BookSources/978-0-547-16702-2"><bdi>978-0-547-16702-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=9th&rft.pub=Brooks%2FCole&rft.date=2009&rft.isbn=978-0-547-16702-2&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaximum+and+minimum" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomasWeirHass2010" class="citation book cs1"><a href="/wiki/George_B._Thomas" title="George B. Thomas">Thomas, George B.</a>; Weir, Maurice D.; <a href="/wiki/Joel_Hass" title="Joel Hass">Hass, Joel</a> (2010). <i>Thomas' Calculus: Early Transcendentals</i> (12th ed.). <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-58876-0" title="Special:BookSources/978-0-321-58876-0"><bdi>978-0-321-58876-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thomas%27+Calculus%3A+Early+Transcendentals&rft.edition=12th&rft.pub=Addison-Wesley&rft.date=2010&rft.isbn=978-0-321-58876-0&rft.aulast=Thomas&rft.aufirst=George+B.&rft.au=Weir%2C+Maurice+D.&rft.au=Hass%2C+Joel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaximum+and+minimum" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Minimum.html">"Minimum"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Minimum&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMinimum.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaximum+and+minimum" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Maximum.html">"Maximum"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Maximum&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMaximum.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaximum+and+minimum" class="Z3988"></span></span> </li> <li id="cite_note-minimization_maximization_refresher-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-minimization_maximization_refresher_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-minimization_maximization_refresher_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarrett,_Paul" class="citation web cs1">Garrett, Paul. <a rel="nofollow" class="external text" href="https://mathinsight.org/minimization_maximization_refresher">"Minimization and maximization refresher"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Minimization+and+maximization+refresher&rft.au=Garrett%2C+Paul&rft_id=https%3A%2F%2Fmathinsight.org%2Fminimization_maximization_refresher&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMaximum+and+minimum" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://physics.usyd.edu.au/teach_res/mp/doc/math_sinc_function.pdf">The Unnormalized Sinc Function</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170215045226/http://www.physics.usyd.edu.au/teach_res/mp/doc/math_sinc_function.pdf">Archived</a> 2017-02-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>", University of Sydney</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">For clarity, we refer to the input (<i>x</i>) as <i>points</i> and the output (<i>y</i>) as <i>values;</i> compare <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a> and <a href="/wiki/Critical_value_(critical_point)" class="mw-redirect" title="Critical value (critical point)">critical value</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Maximum_and_minimum&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/maxima" class="extiw" title="wiktionary:maxima">maxima</a></b></i>, <i><b><a href="https://en.wiktionary.org/wiki/minima" class="extiw" title="wiktionary:minima">minima</a></b></i>, or <i><b><a href="https://en.wiktionary.org/wiki/extremum" class="extiw" title="wiktionary:extremum">extremum</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.maa.org/publications/periodicals/convergence/thomas-simpson-and-maxima-and-minima">Thomas Simpson's work on Maxima and Minima</a> at <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070713083148/http://mathdl.maa.org/convergence/1/">Convergence</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathalino.com/reviewer/differential-calculus/application-of-maxima-and-minima">Application of Maxima and Minima with sub pages of solved problems</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">Jolliffe, Arthur Ernest (1911). <span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Maxima and Minima"><a class="external text" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Maxima_and_Minima">"Maxima and Minima" </a></span>. <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>. 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form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a class="mw-selflink selflink">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Maximum_and_minimum&oldid=1254902629">https://en.wikipedia.org/w/index.php?title=Maximum_and_minimum&oldid=1254902629</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Calculus" title="Category:Calculus">Calculus</a></li><li><a href="/wiki/Category:Mathematical_analysis" title="Category:Mathematical analysis">Mathematical analysis</a></li><li><a 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