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<div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="विकिपीडिया में खोजें" aria-label="विकिपीडिया में खोजें" autocapitalize="sentences" title="इस विकि में खोजें [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="विशेष:खोज"> </div> <button class="cdx-button cdx-search-input__end-button">खोजें</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="व्यक्तिगत उपकरण"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="दिखावट"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="पृष्ठ के फ़ॉन्ट का आकार, चौड़ाई और रंग बदलें" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="दिखावट" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">दिखावट</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hi.wikipedia.org&uselang=hi" class=""><span>दान करें</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE_%E0%A4%AC%E0%A4%A8%E0%A4%BE%E0%A4%8F%E0%A4%81&returnto=%E0%A4%B0%E0%A5%89%E0%A4%B2+%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="हमारा सुझाव है की आप खाता बनाएँ और लॉगिन करें, परन्तु यह अनिवार्य नहीं है" class=""><span>खाता बनाएँ</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A4%BF%E0%A4%A8&returnto=%E0%A4%B0%E0%A5%89%E0%A4%B2+%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="आपको सत्रारम्भ करने के लिए प्रोत्साहित किया जाता है; लेकिन यह अनिवार्य नहीं है [o]" accesskey="o" class=""><span>लॉग-इन करें</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="अन्य विकल्प" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="व्यक्तिगत उपकरण" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">व्यक्तिगत उपकरण</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hi.wikipedia.org&uselang=hi"><span>दान करें</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE_%E0%A4%AC%E0%A4%A8%E0%A4%BE%E0%A4%8F%E0%A4%81&returnto=%E0%A4%B0%E0%A5%89%E0%A4%B2+%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="हमारा सुझाव है की आप खाता बनाएँ और लॉगिन करें, परन्तु यह अनिवार्य नहीं है"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>खाता बनाएँ</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A4%BF%E0%A4%A8&returnto=%E0%A4%B0%E0%A5%89%E0%A4%B2+%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="आपको सत्रारम्भ करने के लिए प्रोत्साहित किया जाता है; लेकिन यह अनिवार्य नहीं है [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>लॉग-इन करें</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> लॉग-आउट किए गए संपादकों के लिए पृष्ठ <a href="/wiki/%E0%A4%B8%E0%A4%B9%E0%A4%BE%E0%A4%AF%E0%A4%A4%E0%A4%BE:%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%9A%E0%A4%AF" aria-label="संपादन के बारे में अधिक जानें"><span>अधिक जानें</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A5%87%E0%A4%B0%E0%A5%87_%E0%A4%AF%E0%A5%8B%E0%A4%97%E0%A4%A6%E0%A4%BE%E0%A4%A8" title="इस आईपी पते से संपादन की सूची [y]" accesskey="y"><span>योगदान</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A5%87%E0%A4%B0%E0%A5%80_%E0%A4%B5%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%BE" title="इस आईपी एड्रेससे हुए बदलावों के बारे में वार्ता [n]" accesskey="n"><span>वार्ता</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="साइट"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="सामग्री" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">सामग्री</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">साइडबार पर जाएँ</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">छुपाएँ</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">शुरुआत</div> </a> </li> <li id="toc-प्रमेय_का_गणितीय_रूप" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#प्रमेय_का_गणितीय_रूप"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>प्रमेय का गणितीय रूप</span> </div> </a> <ul id="toc-प्रमेय_का_गणितीय_रूप-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-सन्दर्भ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#सन्दर्भ"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>सन्दर्भ</span> </div> </a> <ul id="toc-सन्दर्भ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-इन्हें_भी_देखें" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#इन्हें_भी_देखें"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>इन्हें भी देखें</span> </div> </a> <ul id="toc-इन्हें_भी_देखें-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="सामग्री" 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="विषयसूची को टॉगल करें" > <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">विषयसूची को टॉगल करें</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">रॉल प्रमेय</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="दूसरे भाषा में लेख पर जाएँ। 45 भाषाओं में उपलब्ध" > <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-45" 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">45 भाषाएँ</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/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%B1%D9%88%D9%84" title="مبرهنة رول – अरबी" lang="ar" hreflang="ar" data-title="مبرهنة رول" data-language-autonym="العربية" data-language-local-name="अरबी" 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/Roll_teoremi" title="Roll teoremi – अज़रबैजानी" lang="az" hreflang="az" data-title="Roll teoremi" data-language-autonym="Azərbaycanca" data-language-local-name="अज़रबैजानी" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B0%E0%A7%8B%E0%A6%B2%E0%A7%87%E0%A6%B0_%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="রোলের উপপাদ্য – बंगाली" lang="bn" hreflang="bn" data-title="রোলের উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="बंगाली" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Rolle" title="Teorema de Rolle – कातालान" lang="ca" hreflang="ca" data-title="Teorema de Rolle" data-language-autonym="Català" data-language-local-name="कातालान" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%D9%85%DB%8C_%DA%95%DB%86%D9%84" title="تیۆرمی ڕۆل – सोरानी कुर्दिश" lang="ckb" hreflang="ckb" data-title="تیۆرمی ڕۆل" data-language-autonym="کوردی" data-language-local-name="सोरानी कुर्दिश" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rolleova_v%C4%9Bta" title="Rolleova věta – चेक" lang="cs" hreflang="cs" data-title="Rolleova věta" data-language-autonym="Čeština" data-language-local-name="चेक" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Rolles_s%C3%A6tning" title="Rolles sætning – डेनिश" lang="da" hreflang="da" data-title="Rolles sætning" data-language-autonym="Dansk" data-language-local-name="डेनिश" 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/Satz_von_Rolle" title="Satz von Rolle – जर्मन" lang="de" hreflang="de" data-title="Satz von Rolle" data-language-autonym="Deutsch" data-language-local-name="जर्मन" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%A1%CE%BF%CE%BB" title="Θεώρημα Ρολ – यूनानी" lang="el" hreflang="el" data-title="Θεώρημα Ρολ" data-language-autonym="Ελληνικά" data-language-local-name="यूनानी" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Rolle%27s_theorem" title="Rolle's theorem – अंग्रेज़ी" lang="en" hreflang="en" data-title="Rolle's theorem" data-language-autonym="English" data-language-local-name="अंग्रेज़ी" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Teoremo_de_Rolle" title="Teoremo de Rolle – एस्पेरेंतो" lang="eo" hreflang="eo" data-title="Teoremo de Rolle" data-language-autonym="Esperanto" data-language-local-name="एस्पेरेंतो" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_Rolle" title="Teorema de Rolle – स्पेनिश" lang="es" hreflang="es" data-title="Teorema de Rolle" data-language-autonym="Español" data-language-local-name="स्पेनिश" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Rolleren_teorema" title="Rolleren teorema – बास्क" lang="eu" hreflang="eu" data-title="Rolleren teorema" data-language-autonym="Euskara" data-language-local-name="बास्क" 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/%D9%82%D8%B6%DB%8C%D9%87_%D8%B1%D9%88%D9%84" title="قضیه رول – फ़ारसी" lang="fa" hreflang="fa" data-title="قضیه رول" data-language-autonym="فارسی" data-language-local-name="फ़ारसी" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rollen_lause" title="Rollen lause – फ़िनिश" lang="fi" hreflang="fi" data-title="Rollen lause" data-language-autonym="Suomi" data-language-local-name="फ़िनिश" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Rolle" title="Théorème de Rolle – फ़्रेंच" lang="fr" hreflang="fr" data-title="Théorème de Rolle" data-language-autonym="Français" data-language-local-name="फ़्रेंच" 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/Teorema_de_Rolle" title="Teorema de Rolle – गैलिशियन" lang="gl" hreflang="gl" data-title="Teorema de Rolle" data-language-autonym="Galego" data-language-local-name="गैलिशियन" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A8%D7%95%D7%9C" title="משפט רול – हिब्रू" lang="he" hreflang="he" data-title="משפט רול" data-language-autonym="עברית" data-language-local-name="हिब्रू" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Rolleov_teorem" title="Rolleov teorem – क्रोएशियाई" lang="hr" hreflang="hr" data-title="Rolleov teorem" data-language-autonym="Hrvatski" data-language-local-name="क्रोएशियाई" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rolle-t%C3%A9tel" title="Rolle-tétel – हंगेरियाई" lang="hu" hreflang="hu" data-title="Rolle-tétel" data-language-autonym="Magyar" data-language-local-name="हंगेरियाई" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8C%D5%B8%D5%AC%D5%AC%D5%AB_%D5%A9%D5%A5%D5%B8%D6%80%D5%A5%D5%B4" title="Ռոլլի թեորեմ – आर्मेनियाई" lang="hy" hreflang="hy" data-title="Ռոլլի թեորեմ" data-language-autonym="Հայերեն" data-language-local-name="आर्मेनियाई" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_Rolle" title="Teorema Rolle – इंडोनेशियाई" lang="id" hreflang="id" data-title="Teorema Rolle" data-language-autonym="Bahasa Indonesia" data-language-local-name="इंडोनेशियाई" 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/Regla_Rolles" title="Regla Rolles – आइसलैंडिक" lang="is" hreflang="is" data-title="Regla Rolles" data-language-autonym="Íslenska" data-language-local-name="आइसलैंडिक" 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/Teorema_di_Rolle" title="Teorema di Rolle – इतालवी" lang="it" hreflang="it" data-title="Teorema di Rolle" data-language-autonym="Italiano" data-language-local-name="इतालवी" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86" title="ロルの定理 – जापानी" lang="ja" hreflang="ja" data-title="ロルの定理" data-language-autonym="日本語" data-language-local-name="जापानी" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BB%D0%BB%D1%8C_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Ролль теоремасы – कज़ाख़" lang="kk" hreflang="kk" data-title="Ролль теоремасы" data-language-autonym="Қазақша" data-language-local-name="कज़ाख़" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%A4%EC%9D%98_%EC%A0%95%EB%A6%AC" title="롤의 정리 – कोरियाई" lang="ko" hreflang="ko" data-title="롤의 정리" data-language-autonym="한국어" data-language-local-name="कोरियाई" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teorema_de_Rolle" title="Teorema de Rolle – लॉमबर्ड" lang="lmo" hreflang="lmo" data-title="Teorema de Rolle" data-language-autonym="Lombard" data-language-local-name="लॉमबर्ड" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Rolle" title="Stelling van Rolle – डच" lang="nl" hreflang="nl" data-title="Stelling van Rolle" data-language-autonym="Nederlands" data-language-local-name="डच" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rolles_teorem" title="Rolles teorem – नॉर्वेजियाई बोकमाल" lang="nb" hreflang="nb" data-title="Rolles teorem" data-language-autonym="Norsk bokmål" data-language-local-name="नॉर्वेजियाई बोकमाल" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Rolle%E2%80%99a" title="Twierdzenie Rolle’a – पोलिश" lang="pl" hreflang="pl" data-title="Twierdzenie Rolle’a" data-language-autonym="Polski" data-language-local-name="पोलिश" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_%C3%ABd_Rolle" title="Teorema ëd Rolle – Piedmontese" lang="pms" hreflang="pms" data-title="Teorema ëd Rolle" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Rolle" title="Teorema de Rolle – पुर्तगाली" lang="pt" hreflang="pt" data-title="Teorema de Rolle" data-language-autonym="Português" data-language-local-name="पुर्तगाली" 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/Teorema_lui_Rolle" title="Teorema lui Rolle – रोमानियाई" lang="ro" hreflang="ro" data-title="Teorema lui Rolle" data-language-autonym="Română" data-language-local-name="रोमानियाई" 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%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%A0%D0%BE%D0%BB%D0%BB%D1%8F" title="Теорема Ролля – रूसी" lang="ru" hreflang="ru" data-title="Теорема Ролля" data-language-autonym="Русский" data-language-local-name="रूसी" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Rollova_veta_o_strednej_hodnote" title="Rollova veta o strednej hodnote – स्लोवाक" lang="sk" hreflang="sk" data-title="Rollova veta o strednej hodnote" data-language-autonym="Slovenčina" data-language-local-name="स्लोवाक" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teorema_e_Rolle-s" title="Teorema e Rolle-s – अल्बानियाई" lang="sq" hreflang="sq" data-title="Teorema e Rolle-s" data-language-autonym="Shqip" data-language-local-name="अल्बानियाई" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%BE%D0%BB%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Ролова теорема – सर्बियाई" lang="sr" hreflang="sr" data-title="Ролова теорема" data-language-autonym="Српски / srpski" data-language-local-name="सर्बियाई" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rolles_sats" title="Rolles sats – स्वीडिश" lang="sv" hreflang="sv" data-title="Rolles sats" data-language-autonym="Svenska" data-language-local-name="स्वीडिश" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B0%E0%AF%8B%E0%AE%B2%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="ரோலின் தேற்றம் – तमिल" lang="ta" hreflang="ta" data-title="ரோலின் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="तमिल" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%82%E0%B8%A3%E0%B8%A5%E0%B8%A5%E0%B9%8C" title="ทฤษฎีบทของโรลล์ – थाई" lang="th" hreflang="th" data-title="ทฤษฎีบทของโรลล์" data-language-autonym="ไทย" data-language-local-name="थाई" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Rolle_teoremi" title="Rolle teoremi – तुर्की" lang="tr" hreflang="tr" data-title="Rolle teoremi" data-language-autonym="Türkçe" data-language-local-name="तुर्की" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%A0%D0%BE%D0%BB%D0%BB%D1%8F" title="Теорема Ролля – यूक्रेनियाई" lang="uk" hreflang="uk" data-title="Теорема Ролля" data-language-autonym="Українська" data-language-local-name="यूक्रेनियाई" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Rolle" title="Định lý Rolle – वियतनामी" lang="vi" hreflang="vi" data-title="Định lý Rolle" data-language-autonym="Tiếng Việt" data-language-local-name="वियतनामी" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BD%97%E5%B0%94%E5%AE%9A%E7%90%86" title="罗尔定理 – चीनी" lang="zh" hreflang="zh" data-title="罗尔定理" data-language-autonym="中文" data-language-local-name="चीनी" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q193286#sitelinks-wikipedia" title="अंतरभाषा कड़ियाँ संपादित करें" class="wbc-editpage">कड़ियाँ संपादित करें</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="नामस्थान"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a 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data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">दिखावट</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">साइडबार पर जाएँ</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">छुपाएँ</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">मुक्त ज्ञानकोश विकिपीडिया से</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="hi" dir="ltr"><p>कलन विधियों में <b>रॉल प्रमेय</b> अथवा <b>रोल का प्रमेय</b> (Rolle's theorem) के अनुसार यदि किसी <a href="/wiki/%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2%E0%A4%A8" class="mw-redirect" title="अवकलन">अवकलनीय</a> <a href="/wiki/%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="फलन">फलन</a> का मान दो अलग-अलग बिन्दुओं पर समान है तो आवश्यक रूप से उन दोनो बिन्दुओं के मध्य कम से कम एक बिन्दु ऐसा होगा जिसका <a href="/wiki/%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2%E0%A4%9C" title="अवकलज">अवकलज</a> (फलन के ग्राफ में <a href="/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%B6%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE" class="mw-redirect" title="स्पर्शज्या">स्पर्शज्या</a> का ढाल) शून्य अर्थात (०) होगा। रॉल का प्रमेय, <a href="/wiki/%E0%A4%AE%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF%E0%A4%AE%E0%A4%BE%E0%A4%A8_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="माध्यमान प्रमेय">माध्यमान प्रमेय</a> (mean value theorem) का एक विशिष्ट रूप है। रॉल के प्रमेय का उपयोग मध्यमान प्रमेय को सिद्ध करने के लिये किया जाता है। टेलर के प्रमेय को सिद्ध करने का आधार भी यही प्रमेय है। </p> <div class="mw-heading mw-heading2"><h2 id="प्रमेय_का_गणितीय_रूप"><span id=".E0.A4.AA.E0.A5.8D.E0.A4.B0.E0.A4.AE.E0.A5.87.E0.A4.AF_.E0.A4.95.E0.A4.BE_.E0.A4.97.E0.A4.A3.E0.A4.BF.E0.A4.A4.E0.A5.80.E0.A4.AF_.E0.A4.B0.E0.A5.82.E0.A4.AA"></span>प्रमेय का गणितीय रूप</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B0%E0%A5%89%E0%A4%B2_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF&action=edit&section=1" title="अनुभाग सम्पादित करें: प्रमेय का गणितीय रूप"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0:RTCalc.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/RTCalc.svg/300px-RTCalc.svg.png" decoding="async" width="300" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/RTCalc.svg/450px-RTCalc.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/RTCalc.svg/600px-RTCalc.svg.png 2x" data-file-width="734" data-file-height="320" /></a><figcaption>रॉल के प्रमेय का एक सरल उदाहरण</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0:Teorema_de_Rolle_(caso_3).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Teorema_de_Rolle_%28caso_3%29.jpg/300px-Teorema_de_Rolle_%28caso_3%29.jpg" decoding="async" width="300" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Teorema_de_Rolle_%28caso_3%29.jpg/450px-Teorema_de_Rolle_%28caso_3%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/9/9b/Teorema_de_Rolle_%28caso_3%29.jpg 2x" data-file-width="500" data-file-height="250" /></a><figcaption> रॉल के प्रमेय का दूसरा उदाहरण जिसमें खुले अन्तराल (a,b) में दो बिन्दुओं c1 औ c2 पर ग्राफ की प्रवणता शून्य है।</figcaption></figure> <p><a href="/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या">वास्तविक</a> मान वाला फलन <i>f</i> इस प्रकार परिभाषित है कि यह बन्द अन्तराल [a,b] में सतत है, खुले अन्तराल (a,b) में अवकलनीय है और <i>f(a) = f(b)</i> तब ∃ c ∈ (a,b), इस प्रकार होगा की </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'(c)=0.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(c)=0.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48b5e1aeed3f814a1a9bc462bdf79843b4adddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.116ex; height:3.009ex;" alt="{\displaystyle f'(c)=0.\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="सन्दर्भ"><span id=".E0.A4.B8.E0.A4.A8.E0.A5.8D.E0.A4.A6.E0.A4.B0.E0.A5.8D.E0.A4.AD"></span>सन्दर्भ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%B0%E0%A5%89%E0%A4%B2_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF&action=edit&section=2" title="अनुभाग सम्पादित करें: सन्दर्भ"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="इन्हें_भी_देखें"><span 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