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(Osnabrück 2023-2024)/Part I/Lecture 15">Lecture 15</a></bdi></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>\setcounter{section}{15}<br /><br /> </p><p><br /> <br /> <br /> <br /> <br /> \subtitle {Higher derivatives} </p><p>The derivative $f'$ of a differentiable function is also called the \keyword {first derivative} {} of $f$. The zeroth derivative is the function itself. Higher derivatives are defined recursively. <br /><br /><br /> \inputdefinition<br />{ }<br />{ </p> <div> <p>Let <br />\mathrelationchain<br />{\relationchain<br />{ I }<br />{ \subseteq }{ \R }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} denote an <a href="/wiki/Real_numbers/Intervals/Definition" title="Real numbers/Intervals/Definition">interval</a>, and let <br />\mathdisp {f \colon I \longrightarrow \R} { }<br /> be a <a href="/wiki/Mapping/Definition" title="Mapping/Definition">function</a>. The function $f$ is called $n$-times \definitionword {differentiable}{,} if it is \mathl{(n-1)}{-}times differentiable, and the \mathl{(n-1)}{-}th derivative, that is \mathl{f^{(n-1)}}{,} is also <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a>. The derivative <br />\mathrelationchaindisplay<br />{\relationchain<br />{ f^{(n)} (x) }<br />{ \defeq} {(f^{(n-1)})' (x) }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{} </p> is called the $n$-th \definitionword {derivative}{} of $f$. </div><p> } </p><p>The second derivative is written as \mathl{f^{\prime \prime}}{,} the third derivative as \mathl{f^{\prime \prime \prime}}{.} If a function is $n$-times differentiable, then we say that the derivatives exist up to \keyword {order} {} $n$. A function $f$ is called \keyword {infinitely often differentiable} {,} if it is $n$-times differentiable for every $n$. </p><p>A differentiable function is continuous due to <a href="/wiki/Differentiable_function/D_in_R/Continuity_in_point/Fact" title="Differentiable function/D in R/Continuity in point/Fact">Corollary 14.6 </a>, but its derivative is not necessarily so. Therefore, the following concept is justified. <br /><br /><br /> \inputdefinition<br />{ }<br />{ </p> <div> <p>Let <br />\mathrelationchain<br />{\relationchain<br />{I }<br />{ \subseteq }{ \R }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} be an <a href="/wiki/Real_numbers/Intervals/Definition" title="Real numbers/Intervals/Definition">interval</a>, and let <br />\mathdisp {f \colon I \longrightarrow \R} { }<br /> be a <a href="/wiki/Mapping/Definition" title="Mapping/Definition">function</a>. The function $f$ is called \definitionword {continuously differentiable}{,} if $f$ is <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> and its <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">derivative</a> $f'$ is </p> <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous</a>. </div><p> } </p><p>A function is called $n$-times \keyword {continuously differentiable} {,} if it is $n$-times differentiable, and its $n$-th derivative is continuous. </p><p><br /> <br /> <br /> <br /> <br /> \subtitle {Extrema of functions} </p><p>We investigate now, with the help of the methods from differentiability, when a differentiable function <br />\mathdisp {f \colon I \longrightarrow \R} { , }<br /> where <br />\mathrelationchain<br />{\relationchain<br />{I }<br />{ \subseteq }{ \R }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} denotes an interval, has a \extrabracket {local} {} {} extremum, and how the growing behavior looks like. </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Real function/Open interval/Local extrema/Differentiable/Derivative zero/Fact}<br />{Theorem}<br />{}<br />{ <p>\factsituation {Let <br />\mathdisp {f \colon {]a,b[} \longrightarrow \R} { }<br /> be a <a href="/wiki/Mapping/Definition" title="Mapping/Definition">function</a>}<br />\factcondition {which attains in <br />\mathrelationchain<br />{\relationchain<br />{c }<br />{ \in }{ {]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} a <a href="/wiki/Real_function/Local_maximum_and_minimum/Definition" title="Real function/Local maximum and minimum/Definition">local extremum</a>, and is <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> there.} <br />\factconclusion {Then <br />\mathrelationchain<br />{\relationchain<br />{f'(c) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds.}<br />\factextra {}<br /> }<br />{ </p> <div> <div> <p>We may assume that $f$ attains a local maximum in $c$. This means that there exists an <br />\mathrelationchain<br />{\relationchain<br />{ \epsilon }<br />{ > }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} such that <br />\mathrelationchain<br />{\relationchain<br />{f(x) }<br />{ \leq }{f(c) }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds for all <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \in }{ [c - \epsilon, c + \epsilon] }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Let \mathl{{ \left( s_n \right) }_{n \in \N }}{} be a sequence with <br />\mathrelationchain<br />{\relationchain<br />{ c- \epsilon }<br />{ \leq }{s_n }<br />{ < }{ c }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} tending to $c$ \extrabracket {\quotationshort{from below}{}} {} {.} Then <br />\mathrelationchain<br />{\relationchain<br />{ s_n- c }<br />{ < }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and so <br />\mathrelationchain<br />{\relationchain<br />{f(s_n) -f(c) }<br />{ \leq }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and therefore the difference quotient <br />\mathrelationchaindisplay<br />{\relationchain<br />{ \frac{ f (s_n )-f (c) }{ s_n -c } }<br />{ \geq} { 0 }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.} Due to <a href="/wiki/Real_numbers/Convergent_sequences/Compare/Fact" title="Real numbers/Convergent sequences/Compare/Fact">Lemma 7.12 </a>, this relation carries over to the limit, which is the derivative. Hence, <br />\mathrelationchain<br />{\relationchain<br />{f'(c) }<br />{ \geq }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} For another sequence \mathl{{ \left( t_n \right) }_{n \in \N }}{} with <br />\mathrelationchain<br />{\relationchain<br />{ c + \epsilon }<br />{ \geq }{ t_n }<br />{ > }{ c }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} we get <br />\mathrelationchaindisplay<br />{\relationchain<br />{ \frac{ f (t_n )-f (c) }{ t_n -c } }<br />{ \leq} { 0 }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.} Therefore, also <br />\mathrelationchain<br />{\relationchain<br />{f'(c) }<br />{ \leq }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} and thus <br />\mathrelationchain<br />{\relationchain<br />{f'(c) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} </p> </div> </div> }</div> <p><br /> <br /><br /><br /><br /><br />\image{ \begin{center}<br />\includegraphics[width=5.5cm]{\imageinclude {X_Cubed.svg} }<br />\end{center} <br />\imagetext {} }<br /><br />\imagelicense { X Cubed.svg } {} {Pieter Kuiper} {Commons} {PD} {}<br /> <br /> </p><p><br /> We remark that the vanishing of the derivative is only a necessary, but not a sufficient, criterion for the existence of an extremum. The easiest example for this phenomenon is the function $\R \rightarrow \R , x \mapsto x^3$, which is strictly increasing and its derivative is zero at the zero point. We will provide a sufficient criterion in <a href="/wiki/Real_function/Extrema/Second_derivative/Fact" title="Real function/Extrema/Second derivative/Fact">Corollary 15.9 </a> below, see also <a href="/wiki/Real_function/Extrema/Higher_derivatives/Fact" title="Real function/Extrema/Higher derivatives/Fact">Theorem 17.4 </a>. </p><p><br /> <br /> <br /> <br /> <br /> \subtitle {The mean value theorem} </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Real function/Theorem of Rolle/Fact}<br />{Theorem}<br />{}<br />{ <p>\factsituation {Let <br />\mathrelationchain<br />{\relationchain<br />{a }<br />{ < }{b }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and let <br />\mathdisp {f \colon [a,b] \longrightarrow \R} { }<br /> be a <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous</a> function, which is <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> on \mathl{]a,b[}{,}}<br />\factcondition {and such that <br />\mathrelationchain<br />{\relationchain<br />{f(a) }<br />{ = }{f(b) }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.}} <br />\factconclusion {Then there exists some <br />\mathrelationchain<br />{\relationchain<br />{c }<br />{ \in }{ {]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} such that <br />\mathrelationchaindisplay<br />{\relationchain<br />{f'(c) }<br />{ =} { 0 }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.}}<br />\factextra {}<br /> }<br />{ </p> <div> <p>The statement is true if $f$ is constant. So suppose that $f$ is not constant. Then there exists some <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \in }{{]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} such that <br />\mathrelationchain<br />{\relationchain<br />{f(x) }<br />{ \neq }{ f(a) }<br />{ = }{ f(b) }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Let's say that \mathl{f(x)}{} has a larger value. Due to <a href="/wiki/Continuous_function/Closed_bounded_interval/Maximum_is_attained/Fact" title="Continuous function/Closed bounded interval/Maximum is attained/Fact">Theorem 11.13 </a>, there exists some <br />\mathrelationchain<br />{\relationchain<br />{c }<br />{ \in }{ [a,b] }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} where the function attains its <a href="/wiki/Realvalued_function/Set/Maximum/Definition" title="Realvalued function/Set/Maximum/Definition">maximum</a>. This point is not on the border. For this $c$, we have <br />\mathrelationchain<br />{\relationchain<br />{f'(c) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} due to <a href="/wiki/Real_function/Open_interval/Local_extrema/Differentiable/Derivative_zero/Fact" title="Real function/Open interval/Local extrema/Differentiable/Derivative zero/Fact">Theorem 15.3 </a>. </p> </div> }</div> <p><br /> This theorem is called \keyword {Theorem of Rolle} {.} <br /><br /><br /><br /><br />\image{ \begin{center}<br />\includegraphics[width=5.5cm]{\imageinclude {Mvt2_italian.svg} }<br />\end{center} <br />\imagetext {The mean value theorem means that, for every secant, there exists a parallel tangent.} }<br /><br />\imagelicense { Mvt2 italian.svg } {} {4C} {Commons} {CC-by-sa 3.0} {}<br /> <br /> </p><p>The following theorem is called \keyword {Mean value theorem} {.} It says that if a function describes a differentiable one-dimensional movement, then the average velocity is obtained at least once as the instantaneous velocity. </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Differentiable functions/Mean value theorem/Fact}<br />{Theorem}<br />{}<br />{ <p>\factsituation {Let <br />\mathrelationchain<br />{\relationchain<br />{a }<br />{ < }{b }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and let <br />\mathdisp {f \colon [a,b] \longrightarrow \R} { }<br /> be a <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous function</a> which is <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> on \mathl{]a,b[}{.}} <br />\factconclusion {Then there exists some <br />\mathrelationchain<br />{\relationchain<br />{c}<br />{ \in }{{]a,b[}}<br />{ }{}<br />{ }{}<br />{ }{}<br />} {}{}{,} such that <br />\mathrelationchaindisplay<br />{\relationchain<br />{ f'(c)}<br />{ =} { { \frac{ f(b)-f(a) }{ b-a } } }<br />{ } {}<br />{ } {}<br />{ } {}<br />} {}{}{.}}<br />\factextra {}<br /> }<br />{ </p> <div> <p>We consider the auxiliary function <br />\mathdisp {g \colon [a,b] \longrightarrow \R , x \longmapsto g(x) \defeq f(x) - { \frac{ f(b) -f(a) }{ b-a } } (x-a)} { . }<br /> This function is also <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous</a> and <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> in \mathl{]a,b[}{.} Moreover, we have <br />\mathrelationchain<br />{\relationchain<br />{ g(a) }<br />{ = }{ f(a) }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} and <br />\mathrelationchaindisplay<br />{\relationchain<br />{ g(b) }<br />{ =} { f(b) -(f(b)-f(a)) }<br />{ =} { f(a) }<br />{ } { }<br />{ } { }<br />} {}{}{.} Hence, $g$ fulfills the conditions of <a href="/wiki/Real_function/Theorem_of_Rolle/Fact" title="Real function/Theorem of Rolle/Fact">Theorem 15.4 </a>, and therefore there exists some <br />\mathrelationchain<br />{\relationchain<br />{c }<br />{ \in }{ {]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} such that <br />\mathrelationchain<br />{\relationchain<br />{g'(c) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Because of the rules for derivatives, we obtain <br />\mathrelationchaindisplay<br />{\relationchain<br />{ f'(c) }<br />{ =} { { \frac{ f(b) -f(a) }{ b-a } } }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.} </p> </div> }</div> <p><br /> </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Real function/Derivative zero/Constant/Fact}<br />{Corollary}<br />{}<br />{ <p>\factsituation {Let <br />\mathdisp {f \colon { ]a,b[} \longrightarrow \R} { }<br /> be a <a href="/wiki/Real_function/Derivative_function/Definition" title="Real function/Derivative function/Definition">differentiable function</a>}<br />\factcondition {such that <br />\mathrelationchain<br />{\relationchain<br />{ f'(x) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} for all <br />\mathrelationchain<br />{\relationchain<br />{ x }<br />{ \in }{ {]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.}} <br />\factconclusion {Then $f$ is constant.}<br />\factextra {}<br /> }<br />{ </p> <div> <p>If $f$ is not constant, then there exists some <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ < }{x' }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} such that <br />\mathrelationchain<br />{\relationchain<br />{ f(x) }<br />{ \neq }{ f(x') }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Then there exists, due to the <a href="/wiki/Differentiable_functions/Mean_value_theorem/Fact" title="Differentiable functions/Mean value theorem/Fact">mean value theorem</a>, some <br />\mathcond {c} {} <br /> {x <c < x'} {} <br /> {} {} {} {,} such that <br />\mathrelationchain<br />{\relationchain<br />{f'(c) }<br />{ = }{ \frac{f(x') - f(x)}{x'-x} }<br />{ \neq }{ 0 }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} which contradicts the assumption. </p> </div> }</div> <p><br /> </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Real function/Derivative/Monotonicity/Fact}<br />{Theorem}<br />{}<br />{ <p>\factsituation {Let <br />\mathrelationchain<br />{\relationchain<br />{I }<br />{ \subseteq }{ \R }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} be an <a href="/wiki/Real_numbers/Intervals/Definition" title="Real numbers/Intervals/Definition">open interval</a>, and let <br />\mathdisp {f \colon I \longrightarrow \R} { }<br /> be a <a href="/wiki/Real_function/Derivative_function/Definition" title="Real function/Derivative function/Definition">differentiable function</a>.}<br />\factsegue {Then the following statements hold.} <br />\factconclusion {\enumerationthree {The function $f$ is increasing \extrabracket {decreasing} {} {} on $I$, if and only if <br />\mathrelationchain<br />{\relationchain<br />{f'(x) }<br />{ \geq }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} \extrabracket {<br />\mathrelationchain<br />{\relationchain<br />{f'(x) }<br />{ \leq }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{}} {} {} holds for all <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \in }{I }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} } {If <br />\mathrelationchain<br />{\relationchain<br />{f'(x) }<br />{ \geq }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds for all <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \in }{I }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and $f'$ has only finitely many zeroes, then $f$ is strictly increasing. } {If <br />\mathrelationchain<br />{\relationchain<br />{f'(x) }<br />{ \leq }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds for all <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \in }{I }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and $f'$ has only finitely many zeroes, then $f$ is strictly decreasing.}}<br />\factextra {}<br /> }<br />{ </p> <div> <p>(1). It is enough to prove the statements for increasing functions. If $f$ is increasing and <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \in }{I }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} then the <a href="/wiki/Difference_quotient/D_in_R/Definition" title="Difference quotient/D in R/Definition">difference quotient</a> fulfills <br />\mathrelationchaindisplay<br />{\relationchain<br />{ \frac{f(x+h) -f(x) }{h} }<br />{ \geq} { 0 }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{} for every $h$ with <br />\mathrelationchain<br />{\relationchain<br />{x+h }<br />{ \in }{I }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} This estimate carries over to the limit as \mathl{h \rightarrow 0}{,} and this limit is \mathl{f'(x)}{.}<br /> Suppose now that the derivative is $\geq 0$. We assume, in order to obtain a contradiction, that there exist two points <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ < }{x' }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} in $I$ with <br />\mathrelationchain<br />{\relationchain<br />{f(x) }<br />{ > }{f(x') }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Due to the <a href="/wiki/Differentiable_functions/Mean_value_theorem/Fact" title="Differentiable functions/Mean value theorem/Fact">mean value theorem</a>, there exists some $c$ with <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ < }{c }<br />{ < }{x' }<br />{ }{ }<br />{ }{ }<br />} {}{}{} and <br />\mathrelationchaindisplay<br />{\relationchain<br />{f'(c) }<br />{ =} {\frac{f(x') - f(x)}{x'-x} }<br />{ <} {0 }<br />{ } { }<br />{ } { }<br />} {}{}{,} which contradicts the condition.<br /> (2). Suppose now that <br />\mathrelationchain<br />{\relationchain<br />{f'(x) }<br />{ > }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds with finitely many exceptions. We assume that <br />\mathrelationchain<br />{\relationchain<br />{f(x) }<br />{ = }{f(x') }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds for two points <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ < }{x' }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Since $f$ is increasing, due to the first part, it follows that $f$ is constant on the interval \mathl{[x,x']}{.} But then <br />\mathrelationchain<br />{\relationchain<br />{f' }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} on this interval, which contradicts the condition that $f'$ has only finitely many zeroes.<br /> </p> </div> }</div> <p><br /> </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Polynomial function/Function behavior from differentiability/Fact}<br />{Corollary}<br />{}<br />{ <p>A real <a href="/wiki/Polynomial/Field/1/Definition" title="Polynomial/Field/1/Definition">polynomial function</a> <br />\mathdisp {f \colon \R \longrightarrow \R} { }<br /> of <a href="/wiki/Polynomial/1/Degree/Definition" title="Polynomial/1/Degree/Definition">degree</a> <br />\mathrelationchain<br />{\relationchain<br />{ d }<br />{ \geq }{ 1 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} has at most \mathl{d-1}{} <a href="/wiki/Real_function/Local_maximum_and_minimum/Definition" title="Real function/Local maximum and minimum/Definition">local extrema</a>, and one can partition the real numbers into at most $d$ intervals, on which $f$ is alternatingly <a href="/wiki/Real_function/Strictly_increasing/Definition" title="Real function/Strictly increasing/Definition">strictly increasing</a> or <a href="/wiki/Real_function/Strictly_decreasing/Definition" title="Real function/Strictly decreasing/Definition">strictly decreasing</a>. </p> }<br />{See <a href="/wiki/Polynomial_function/Function_behavior_from_differentiability/Fact/Proof/Exercise" title="Polynomial function/Function behavior from differentiability/Fact/Proof/Exercise">Exercise 15.14 </a>.}</div> <p><br /> </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Real function/Extrema/Second derivative/Fact}<br />{Corollary}<br />{}<br />{ <p>\factsituation {Let $I$ denote a <a href="/wiki/Real_numbers/Intervals/Definition" title="Real numbers/Intervals/Definition">real interval</a>, <br />\mathdisp {f \colon I \longrightarrow \R} { }<br /> a twice <a href="/wiki/Continuously_differentiable/R/Definition" title="Continuously differentiable/R/Definition">continuously differentiable</a> <a href="/wiki/Mapping/Definition" title="Mapping/Definition">function</a>, and <br />\mathrelationchain<br />{\relationchain<br />{a }<br />{ \in }{I }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} an inner point of the interval.}<br />\factcondition {Suppose that <br />\mathrelationchain<br />{\relationchain<br />{ f'(a) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds.}<br />\factsegue {Then the following statements hold.} <br />\factconclusion {\enumerationtwo {If <br />\mathrelationchain<br />{\relationchain<br />{f^{\prime \prime }(a) }<br />{ > }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds, then $f$ has an isolated local minimum in $a$. } {If <br />\mathrelationchain<br />{\relationchain<br />{ f^{ \prime \prime}(a) }<br />{ < }{0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds, then $f$ has an isolated local maximum in $a$. }}<br />\factextra {}<br /> </p> }<br />{See <a href="/wiki/Real_function/Extrema/Second_derivative/Fact/Proof/Exercise" title="Real function/Extrema/Second derivative/Fact/Proof/Exercise">Exercise 15.15 </a>.}</div> <p><br /> We will encounter a more general statement in <a href="/wiki/Real_function/Extrema/Higher_derivatives/Fact" title="Real function/Extrema/Higher derivatives/Fact">Theorem 17.4 </a>. </p><p><br /> <br /> <br /> <br /> <br /> \subtitle {General mean value theorem and L'Hôpital's rule} </p><p>The following statement is called also the \keyword {general mean value theorem} {.} </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Differentiable function/Mean value theorem/Quotient version/Fact}<br />{Theorem}<br />{}<br />{ <p>\factsituation {Let <br />\mathrelationchain<br />{\relationchain<br />{b }<br />{ > }{a }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and suppose that <br />\mathdisp {f,g \colon [a,b] \longrightarrow \R} { }<br /> are <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous</a> functions which are <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> on \mathl{]a,b[}{} and such that <br />\mathrelationchaindisplay<br />{\relationchain<br />{ g'(x) }<br />{ \neq} {0 }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{} for all <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \in }{{]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.}} <br />\factconclusion {Then <br />\mathrelationchain<br />{\relationchain<br />{ g(b) }<br />{ \neq }{ g(a) }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and there exists some <br />\mathrelationchain<br />{\relationchain<br />{c }<br />{ \in }{{]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} such that <br />\mathrelationchaindisplay<br />{\relationchain<br />{ \frac{f(b)-f(a)}{g(b)-g(a)} }<br />{ =} {\frac{f'(c)}{g'(c)} }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.}}<br />\factextra {}<br /> }<br />{ </p> <div> <p>The statement <br />\mathrelationchaindisplay<br />{\relationchain<br />{g(a) }<br />{ \neq} {g(b) }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{} follows from <a href="/wiki/Real_function/Theorem_of_Rolle/Fact" title="Real function/Theorem of Rolle/Fact">Theorem 15.4 </a>. We consider the auxiliary function <br />\mathrelationchaindisplay<br />{\relationchain<br />{ h(x) }<br />{ \defeq} { f(x)- { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(x) }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.} We have <br />\mathrelationchainalign<br />{\relationchainalign<br />{ h(a) }<br />{ =} { f(a)- { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(a) }<br />{ =} { { \frac{ f(a) g(b) - f(a)g(a) -f(b)g(a)+f(a)g(a) }{ g(b)-g(a) } } }<br />{ =} { { \frac{ f(a) g(b)-f(b)g(a) }{ g(b)-g(a) } } }<br />{ =} { { \frac{ f(b)g(b) - f(b) g(a)-f(b)g(b)+f(a)g(b) }{ g(b)-g(a) } } }<br />} {<br />\relationchainextensionalign<br />{ =} { f(b) - { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(b) }<br />{ =} { h(b) }<br />{ } {}<br />{ } {}<br />} {}{.} Therefore, <br />\mathrelationchain<br />{\relationchain<br />{h(a) }<br />{ = }{h(b) }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and <a href="/wiki/Real_function/Theorem_of_Rolle/Fact" title="Real function/Theorem of Rolle/Fact">Theorem 15.4 </a> yields the existence of some <br />\mathrelationchain<br />{\relationchain<br />{c }<br />{ \in }{{]a,b[} }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} with <br />\mathrelationchaindisplay<br />{\relationchain<br />{h'(c) }<br />{ =} { 0 }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.} Rearranging proves the claim. </p> </div> }</div> <p><br /> From this version, one can recover the mean value theorem, by taking for $g$ the identity. </p><p><br /><br /><br /><br /><br />\image{ \begin{center}<br />\includegraphics[width=5.5cm]{\imageinclude {Guillaume_de_lHopital.jpg} }<br />\end{center} <br />\imagetext {<a href="https://en.wikipedia.org/wiki/Guillaume_de_l%27H%C3%B4pital" class="extiw" title="w:Guillaume de l'Hôpital"> L’Hospital (1661-1704)</a>} }<br /><br />\imagelicense { Guillaume de l'Hôpital.jpg } {} {Bemoeial} {Commons} {PD} {}<br /> <br /> </p><p>For the computation of the limit of a function, the following method called \keyword {L'Hôpital's rule} {} helps. </p> <div class="latex"><br /><br /><br />\inputfactproof<br />{Hospital/Differentiable in inner interval/Fact}<br />{Corollary}<br />{}<br />{ <p>\factsituation {Let <br />\mathrelationchain<br />{\relationchain<br />{I }<br />{ \subseteq }{ \R }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} denote an <a href="/wiki/Real_numbers/Intervals/Definition" title="Real numbers/Intervals/Definition">open interval</a>, and let <br />\mathrelationchain<br />{\relationchain<br />{ a }<br />{ \in }{ I }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} denote a point. Suppose that <br />\mathdisp {f,g \colon I \longrightarrow \R} { }<br /> are <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous functions</a>,}<br />\factcondition {which are <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> on \mathl{I \setminus \{ a \}}{,} fulfilling <br />\mathrelationchain<br />{\relationchain<br />{ f( a ) }<br />{ = }{ g( a ) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} and with <br />\mathrelationchain<br />{\relationchain<br />{ g'(x) }<br />{ \neq }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} for <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ \neq }{a }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Moreover, suppose that the <a href="/wiki/Function/R/Limit/Epsilon/Definition" title="Function/R/Limit/Epsilon/Definition">limit</a> <br />\mathrelationchaindisplay<br />{\relationchain<br />{w }<br />{ \defeq} { \operatorname{lim}_{ x \rightarrow a } \, \frac{f'(x)}{g'(x)} }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{} exists.} <br />\factconclusion {Then also the limit <br />\mathdisp {\operatorname{lim}_{ x \rightarrow a } \, \frac{f(x)}{g(x)}} { }<br /> exists, and it also equals $w$.}<br />\factextra {}<br /> }<br />{ </p> <div> <p>Because $g'$ has no zero in the interval and <br />\mathrelationchain<br />{\relationchain<br />{ g(a) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} holds, it follows, because of <a href="/wiki/Real_function/Theorem_of_Rolle/Fact" title="Real function/Theorem of Rolle/Fact">Theorem 15.4 </a>, that $a$ is the only zero of $g$. Let \mathl{{ \left( x_n \right) }_{n \in \N }}{} denote a <a href="/wiki/Real_numbers/Sequence/Definition" title="Real numbers/Sequence/Definition">sequence</a> in \mathl{I \setminus \{ a \}}{,} <a href="/wiki/Real_numbers/Sequence/Limit_and_convergence/Definition" title="Real numbers/Sequence/Limit and convergence/Definition">converging</a> to $a$. </p><p>For every $x_n$ there exists, due to <a href="/wiki/Differentiable_function/Mean_value_theorem/Quotient_version/Fact" title="Differentiable function/Mean value theorem/Quotient version/Fact">Theorem 15.10 </a>, applied to the interval \mathcor {} {I_n \defeq [x_n, a ]} {or} {[ a ,x_n]} {,} a $c_n$ </p> \extrabracket {in the interior\extrafootnote {<div> <p>The \definitionword {interior}{} of a <a href="/wiki/Real_numbers/Intervals/Definition" title="Real numbers/Intervals/Definition">real interval</a> <br />\mathrelationchain<br />{\relationchain<br />{ I }<br />{ \subseteq }{ \R }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{} </p> is the interval without the boundaries. </div>} {} {} <p>of $I_n$,} {} {} fulfilling <br />\mathrelationchaindisplay<br />{\relationchain<br />{ \frac{f(x_n)-f( a )}{g( x_n )-g( a ) } }<br />{ =} { \frac{f'(c_n)}{g'(c_n)} }<br />{ } { }<br />{ } { }<br />{ } { }<br />} {}{}{.} The sequence \mathl{{ \left( c_n \right) }_{n \in \N }}{} converges also to $a$, so that, because of the condition, the right-hand side converges to <br />\mathrelationchain<br />{\relationchain<br />{ \frac{f'( a )}{g'( a )} }<br />{ = }{ w }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} Therefore, also the left-hand side converges to $w$, and, because of <br />\mathrelationchain<br />{\relationchain<br />{ f( a ) }<br />{ = }{ g( a ) }<br />{ = }{ 0 }<br />{ }{ }<br />{ }{ }<br />} {}{}{,} this means that \mathl{\frac{f(x_n)}{g(x_n)}}{} converges to $w$. </p> </div> }</div> <p><br /> <br /><br /><br />\inputexample{}<br />{ </p> <div> <p>The <a href="/wiki/Polynomial/Field/1/Definition" title="Polynomial/Field/1/Definition">polynomials</a> <br />\mathdisp {3x^2-5x-2 \text{ and } x^3-4x^2+x+6} { }<br /> have both a zero for <br />\mathrelationchain<br />{\relationchain<br />{x }<br />{ = }{2 }<br />{ }{ }<br />{ }{ }<br />{ }{ }<br />} {}{}{.} It is therefore not immediately clear whether the limit <br />\mathdisp {\operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 3x^2-5x-2}{x^3-4x^2+x+6}} { }<br /> exists. Applying twice <a href="/wiki/Hospital/Differentiable_in_inner_interval/Fact" title="Hospital/Differentiable in inner interval/Fact">L'Hôpital's rule</a>, we get the existence and <br />\mathrelationchaindisplay<br />{\relationchain<br />{ \operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 3x^2-5x-2}{x^3-4x^2+x+6} }<br />{ =} { \operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 6x-5}{3x^2-8x+1} }<br />{ =} { \frac{7}{-3} }<br />{ =} { - \frac{7}{3} }<br />{ } { }<br />} {}{}{.} </p> </div><p> }<br /></p>    </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.wikiversity.org/w/index.php?title=Mathematics_for_Applied_Sciences_(Osnabrück_2023-2024)/Part_I/Lecture_15/latex&oldid=2540849">https://en.wikiversity.org/w/index.php?title=Mathematics_for_Applied_Sciences_(Osnabrück_2023-2024)/Part_I/Lecture_15/latex&oldid=2540849</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Category</a>: <ul><li><a href="/wiki/Category:Latexpage" title="Category:Latexpage">Latexpage</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 July 2023, at 14:10.</li> <li id="footer-info-copyright">Text is available under the <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike License</a>; additional terms may apply. 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