# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 15/latex

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\zwischenueberschrift{Higher derivatives}
The derivative $f'$ of a differentiable function is also called the \stichwort {first derivative} of $f$. The zeroth derivative is the function itself. Higher derivatives are defined recursively. \inputdefinition { } {
Let \mavergleichskette {\vergleichskette { I } { \subseteq }{ \R } { }{ } { }{ } { }{ } } denote an interval, and let \mathdisp {f \colon I \longrightarrow \R} { } be a function. The function $f$ is called $n$-times \definitionswort {differentiable}{,} if it is \mathl{(n-1)}{-}times differentiable, and the \mathl{(n-1)}{-}th derivative, that is \mathl{f^{(n-1)}}{,} is also differentiable. The derivative \mavergleichskettedisp {\vergleichskette { f^{(n)} (x) } { \defeq} {(f^{(n-1)})' (x) } { } { } { } { } { } { } }
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 \stichwort {order} $n$. A function $f$ is called \stichwort {infinitely often differentiable} {,} if it is $n$-times differentiable for every $n$.
A differentiable function is continuous due to Corollary 14.6 , but its derivative is not necessarily so. Therefore, the following concept is justified. \inputdefinition { } {
Let \mavergleichskette {\vergleichskette {I } { \subseteq }{ \R } { }{ } { }{ } { }{ } } be an interval, and let \mathdisp {f \colon I \longrightarrow \R} { } be a function. The function $f$ is called \definitionswort {continuously differentiable}{,} if $f$ is differentiable and its derivative $f'$ is
A function is called $n$-times \stichwort {continuously differentiable} {,} if it is $n$-times differentiable, and its $n$-th derivative is continuous.
\zwischenueberschrift{Extrema of functions}
We investigate now, with the help of the methods from differentiability, when a differentiable function \mathdisp {f \colon I \longrightarrow \R} { , } where \mavergleichskette {\vergleichskette {I } { \subseteq }{ \R } { }{ } { }{ } { }{ } } denotes an interval, has a \zusatzklammer {local} extremum, and how the growing behavior looks like.
\faktsituation {Let \mathdisp {f \colona,b \longrightarrow \R} { } be a function} \faktvoraussetzung {which attains in \mavergleichskette {\vergleichskette {c } { \in }{a,b } { }{ } { }{ } { }{ } } a local extremum, and is differentiable there.} \faktfolgerung {Then \mavergleichskette {\vergleichskette {f'(c) } { = }{ 0 } { }{ } { }{ } { }{ } } holds.} \faktzusatz } {
We may assume that $f$ attains a local maximum in $c$. This means that there exists an \mavergleichskette {\vergleichskette { \epsilon } { > }{0 } { }{ } { }{ } { }{ } }{,} such that \mavergleichskette {\vergleichskette {f(x) } { \leq }{f(c) } { }{ } { }{ } { }{ } } holds for all \mavergleichskette {\vergleichskette {x } { \in }{ [c - \epsilon, c + \epsilon] } { }{ } { }{ } { }{ } }{.} Let \mathl{{ \left( s_n \right) }_{n \in \N }} be a sequence with \mavergleichskette {\vergleichskette { c- \epsilon } { \leq }{s_n } { < }{ c } { }{ } { }{ } }{,} tending to $c$ \zusatzklammer {\anfuehrung{from below}} {.} Then \mavergleichskette {\vergleichskette { s_n- c } { < }{0 } { }{ } { }{ } { }{ } }{,} and so \mavergleichskette {\vergleichskette {f(s_n) -f(c) } { \leq }{0 } { }{ } { }{ } { }{ } }{,} and therefore the difference quotient \mavergleichskettedisp {\vergleichskette { \frac{ f (s_n )-f (c) }{ s_n -c } } { \geq} { 0 } { } { } { } { } { } { } }{.} Due to Lemma 7.12 , this relation carries over to the limit, which is the derivative. Hence, \mavergleichskette {\vergleichskette {f'(c) } { \geq }{ 0 } { }{ } { }{ } { }{ } }{.} For another sequence \mathl{{ \left( t_n \right) }_{n \in \N }} with \mavergleichskette {\vergleichskette { c + \epsilon } { \geq }{ t_n } { > }{ c } { }{ } { }{ } }{,} we get \mavergleichskettedisp {\vergleichskette { \frac{ f (t_n )-f (c) }{ t_n -c } } { \leq} { 0 } { } { } { } { } { } { } }{.} Therefore, also \mavergleichskette {\vergleichskette {f'(c) } { \leq }{ 0 } { }{ } { }{ } { }{ } } and thus \mavergleichskette {\vergleichskette {f'(c) } { = }{ 0 } { }{ } { }{ } { }{ } }{.}
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 Corollary 15.9  below, see also Theorem 17.4 .
\zwischenueberschrift{The mean value theorem}
\faktsituation {Let \mavergleichskette {\vergleichskette {a } { < }{b } { }{ } { }{ } { }{ } }{,} and let \mathdisp {f \colon [a,b] \longrightarrow \R} { } be a continuous function, which is differentiable on \mathla,b{,}} \faktvoraussetzung {and such that \mavergleichskette {\vergleichskette {f(a) } { = }{f(b) } { }{ } { }{ } { }{ } }{.}} \faktfolgerung {Then there exists some \mavergleichskette {\vergleichskette {c } { \in }{a,b } { }{ } { }{ } { }{ } }{,} such that \mavergleichskettedisp {\vergleichskette {f'(c) } { =} { 0 } { } { } { } { } { } { } }{.}} \faktzusatz } {
The statement is true if $f$ is constant. So suppose that $f$ is not constant. Then there exists some \mavergleichskette {\vergleichskette {x } { \in }{a,b } { }{ } { }{ } { }{ } }{,} such that \mavergleichskette {\vergleichskette {f(x) } { \neq }{ f(a) } { = }{ f(b) } { }{ } { }{ } }{.} Let's say that \mathl{f(x)} has a larger value. Due to Theorem 11.13 , there exists some \mavergleichskette {\vergleichskette {c } { \in }{ [a,b] } { }{ } { }{ } { }{ } }{,} where the function attains its maximum. This point is not on the border. For this $c$, we have \mavergleichskette {\vergleichskette {f'(c) } { = }{ 0 } { }{ } { }{ } { }{ } }{,} due to Theorem 15.3 .
This theorem is called \stichwort {Theorem of Rolle} {.}
The following theorem is called \stichwort {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.
\faktsituation {Let \mavergleichskette {\vergleichskette {a } { < }{b } { }{ } { }{ } { }{ } }{,} and let \mathdisp {f \colon [a,b] \longrightarrow \R} { } be a continuous function which is differentiable on \mathla,b{.}} \faktfolgerung {Then there exists some \mavergleichskette {\vergleichskette {c} { \in }{a,b} { } { } { } }{,} such that \mavergleichskettedisp {\vergleichskette { f'(c)} { =} { { \frac{ f(b)-f(a) }{ b-a } } } { } { } { } }{.}} \faktzusatz } {
We consider the auxiliary function \mathdisp {g \colon [a,b] \longrightarrow \R , x \longmapsto g(x) \defeq f(x) - { \frac{ f(b) -f(a) }{ b-a } } (x-a)} { . } This function is also continuous and differentiable in \mathla,b{.} Moreover, we have \mavergleichskette {\vergleichskette { g(a) } { = }{ f(a) } { }{ } { }{ } { }{ } } and \mavergleichskettedisp {\vergleichskette { g(b) } { =} { f(b) -(f(b)-f(a)) } { =} { f(a) } { } { } { } { } }{.} Hence, $g$ fulfills the conditions of Theorem 15.4 , and therefore there exists some \mavergleichskette {\vergleichskette {c } { \in }{a,b } { }{ } { }{ } { }{ } }{,} such that \mavergleichskette {\vergleichskette {g'(c) } { = }{ 0 } { }{ } { }{ } { }{ } }{.} Because of the rules for derivatives, we obtain \mavergleichskettedisp {\vergleichskette { f'(c) } { =} { { \frac{ f(b) -f(a) }{ b-a } } } { } { } { } { } { } { } }{.}
\faktsituation {Let \mathdisp {f \colon { ]a,b \longrightarrow \R} { } be a differentiable function} \faktvoraussetzung {such that \mavergleichskette {\vergleichskette { f'(x) } { = }{ 0 } { }{ } { }{ } { }{ } } for all \mavergleichskette {\vergleichskette { x } { \in }{a,b } { }{ } { }{ } { }{ } }{.}} \faktfolgerung {Then $f$ is constant.} \faktzusatz } {
If $f$ is not constant, then there exists some \mavergleichskette {\vergleichskette {x } { < }{x' } { }{ } { }{ } { }{ } } such that \mavergleichskette {\vergleichskette { f(x) } { \neq }{ f(x') } { }{ } { }{ } { }{ } }{.} Then there exists, due to the mean value theorem, some \mathbed {c} {x <c < x'} {,} such that \mavergleichskette {\vergleichskette {f'(c) } { = }{ \frac{f(x') - f(x)}{x'-x} } { \neq }{ 0 } { }{ } { }{ } }{,} which contradicts the assumption.
\faktsituation {Let \mavergleichskette {\vergleichskette {I } { \subseteq }{ \R } { }{ } { }{ } { }{ } } be an open interval, and let \mathdisp {f \colon I \longrightarrow \R} { } be a differentiable function.} \faktuebergang {Then the following statements hold.} \faktfolgerung {\aufzaehlungdrei {The function $f$ is increasing \zusatzklammer {decreasing} on $I$, if and only if \mavergleichskette {\vergleichskette {f'(x) } { \geq }{0 } { }{ } { }{ } { }{ } } \zusatzklammer { \mavergleichskette {\vergleichskette {f'(x) } { \leq }{0 } { }{ } { }{ } { }{ } }} holds for all \mavergleichskette {\vergleichskette {x } { \in }{I } { }{ } { }{ } { }{ } }{.} } {If \mavergleichskette {\vergleichskette {f'(x) } { \geq }{0 } { }{ } { }{ } { }{ } } holds for all \mavergleichskette {\vergleichskette {x } { \in }{I } { }{ } { }{ } { }{ } }{,} and $f'$ has only finitely many zeroes, then $f$ is strictly increasing. } {If \mavergleichskette {\vergleichskette {f'(x) } { \leq }{0 } { }{ } { }{ } { }{ } } holds for all \mavergleichskette {\vergleichskette {x } { \in }{I } { }{ } { }{ } { }{ } }{,} and $f'$ has only finitely many zeroes, then $f$ is strictly decreasing.}} \faktzusatz } {
(1). It is enough to prove the statements for increasing functions. If $f$ is increasing and \mavergleichskette {\vergleichskette {x } { \in }{I } { }{ } { }{ } { }{ } }{,} then the difference quotient fulfills \mavergleichskettedisp {\vergleichskette { \frac{f(x+h) -f(x) }{h} } { \geq} { 0 } { } { } { } { } { } { } } for every $h$ with \mavergleichskette {\vergleichskette {x+h } { \in }{I } { }{ } { }{ } { }{ } }{.} This estimate carries over to the limit as \mathl{h \rightarrow 0}{,} and this limit is \mathl{f'(x)}{.} Suppose now that the derivative is $\geq 0$. We assume, in order to obtain a contradiction, that there exist two points \mavergleichskette {\vergleichskette {x } { < }{x' } { }{ } { }{ } { }{ } } in $I$ with \mavergleichskette {\vergleichskette {f(x) } { > }{f(x') } { }{ } { }{ } { }{ } }{.} Due to the mean value theorem, there exists some $c$ with \mavergleichskette {\vergleichskette {x } { < }{c } { < }{x' } { }{ } { }{ } } and \mavergleichskettedisp {\vergleichskette {f'(c) } { =} {\frac{f(x') - f(x)}{x'-x} } { <} {0 } { } { } { } { } }{,} which contradicts the condition. (2). Suppose now that \mavergleichskette {\vergleichskette {f'(x) } { > }{0 } { }{ } { }{ } { }{ } } holds with finitely many exceptions. We assume that \mavergleichskette {\vergleichskette {f(x) } { = }{f(x') } { }{ } { }{ } { }{ } } holds for two points \mavergleichskette {\vergleichskette {x } { < }{x' } { }{ } { }{ } { }{ } }{.} Since $f$ is increasing, due to the first part, it follows that $f$ is constant on the interval \mathl{[x,x']}{.} But then \mavergleichskette {\vergleichskette {f' } { = }{ 0 } { }{ } { }{ } { }{ } } on this interval, which contradicts the condition that $f'$ has only finitely many zeroes.
A real polynomial function \mathdisp {f \colon \R \longrightarrow \R} { } of degree \mavergleichskette {\vergleichskette { d } { \geq }{ 1 } { }{ } { }{ } { }{ } } has at most \mathl{d-1} local extrema, and one can partition the real numbers into at most $d$ intervals, on which $f$ is alternatingly strictly increasing or strictly decreasing.
\faktsituation {Let $I$ denote a real interval, \mathdisp {f \colon I \longrightarrow \R} { } a twice continuously differentiable function, and \mavergleichskette {\vergleichskette {a } { \in }{I } { }{ } { }{ } { }{ } } an inner point of the interval.} \faktvoraussetzung {Suppose that \mavergleichskette {\vergleichskette { f'(a) } { = }{ 0 } { }{ } { }{ } { }{ } } holds.} \faktuebergang {Then the following statements hold.} \faktfolgerung {\aufzaehlungzwei {If \mavergleichskette {\vergleichskette {f^{\prime \prime }(a) } { > }{ 0 } { }{ } { }{ } { }{ } } holds, then $f$ has an isolated local minimum in $a$. } {If \mavergleichskette {\vergleichskette { f^{ \prime \prime}(a) } { < }{0 } { }{ } { }{ } { }{ } } holds, then $f$ has an isolated local maximum in $a$. }} \faktzusatz
We will encounter a more general statement in Theorem 17.4 .
\zwischenueberschrift{General mean value theorem and L'Hôpital's rule}
The following statement is called also the \stichwort {general mean value theorem} {.}
\faktsituation {Let \mavergleichskette {\vergleichskette {b } { > }{a } { }{ } { }{ } { }{ } }{,} and suppose that \mathdisp {f,g \colon [a,b] \longrightarrow \R} { } are continuous functions which are differentiable on \mathla,b and such that \mavergleichskettedisp {\vergleichskette { g'(x) } { \neq} {0 } { } { } { } { } { } { } } for all \mavergleichskette {\vergleichskette {x } { \in }{a,b } { }{ } { }{ } { }{ } }{.}} \faktfolgerung {Then \mavergleichskette {\vergleichskette { g(b) } { \neq }{ g(a) } { }{ } { }{ } { }{ } }{,} and there exists some \mavergleichskette {\vergleichskette {c } { \in }{a,b } { }{ } { }{ } { }{ } } such that \mavergleichskettedisp {\vergleichskette { \frac{f(b)-f(a)}{g(b)-g(a)} } { =} {\frac{f'(c)}{g'(c)} } { } { } { } { } { } { } }{.}} \faktzusatz } {
The statement \mavergleichskettedisp {\vergleichskette {g(a) } { \neq} {g(b) } { } { } { } { } { } { } } follows from Theorem 15.4 . We consider the auxiliary function \mavergleichskettedisp {\vergleichskette { h(x) } { \defeq} { f(x)- { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(x) } { } { } { } { } { } { } }{.} We have \mavergleichskettealign {\vergleichskettealign { h(a) } { =} { f(a)- { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(a) } { =} { { \frac{ f(a) g(b) - f(a)g(a) -f(b)g(a)+f(a)g(a) }{ g(b)-g(a) } }
} { =} { { \frac{ f(a) g(b)-f(b)g(a) }{ g(b)-g(a) } }
} { =} { { \frac{ f(b)g(b) - f(b) g(a)-f(b)g(b)+f(a)g(b) }{ g(b)-g(a) } }
} } { \vergleichskettefortsetzungalign { =} { f(b) - { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(b) } { =} { h(b) } { } { } }{.} Therefore, \mavergleichskette {\vergleichskette {h(a) } { = }{h(b) } { }{ } { }{ } { }{ } }{,} and Theorem 15.4
yields the existence of some \mavergleichskette {\vergleichskette {c } { \in }{a,b } { }{ } { }{ } { }{ } } with \mavergleichskettedisp {\vergleichskette {h'(c) } { =} { 0 } { } { } { } { } { } { } }{.} Rearranging proves the claim.
From this version, one can recover the mean value theorem, by taking for $g$ the identity.
For the computation of the limit of a function, the following method called \stichwort {L'Hôpital's rule} helps.
\faktsituation {Let \mavergleichskette {\vergleichskette {I } { \subseteq }{ \R } { }{ } { }{ } { }{ } } denote an open interval, and let \mavergleichskette {\vergleichskette { a } { \in }{ I } { }{ } { }{ } { }{ } } denote a point. Suppose that \mathdisp {f,g \colon I \longrightarrow \R} { } are continuous functions,} \faktvoraussetzung {which are differentiable on \mathl{I \setminus \{ a \}}{,} fulfilling \mavergleichskette {\vergleichskette { f( a ) } { = }{ g( a ) } { = }{ 0 } { }{ } { }{ } }{,} and with \mavergleichskette {\vergleichskette { g'(x) } { \neq }{ 0 } { }{ } { }{ } { }{ } } for \mavergleichskette {\vergleichskette {x } { \neq }{a } { }{ } { }{ } { }{ } }{.} Moreover, suppose that the limit \mavergleichskettedisp {\vergleichskette {w } { \defeq} { \operatorname{lim}_{ x \rightarrow a } \, \frac{f'(x)}{g'(x)} } { } { } { } { } { } { } } exists.} \faktfolgerung {Then also the limit \mathdisp {\operatorname{lim}_{ x \rightarrow a } \, \frac{f(x)}{g(x)}} { } exists, and it also equals $w$.} \faktzusatz } {
Because $g'$ has no zero in the interval and \mavergleichskette {\vergleichskette { g(a) } { = }{ 0 } { }{ } { }{ } { }{ } } holds, it follows, because of Theorem 15.4 , that $a$ is the only zero of $g$. Let \mathl{{ \left( x_n \right) }_{n \in \N }} denote a sequence in \mathl{I \setminus \{ a \}}{,} converging to $a$.
For every $x_n$ there exists, due to Theorem 15.10 , applied to the interval \mathkor {I_n \defeq [x_n, a ]} {or} {[ a ,x_n]} {,} a $c_n$
The \definitionswort {interior} of a real interval \mavergleichskette {\vergleichskette { I } { \subseteq }{ \R } { }{ } { }{ } { }{ } }
of $I_n$,} fulfilling \mavergleichskettedisp {\vergleichskette { \frac{f(x_n)-f( a )}{g( x_n )-g( a ) } } { =} { \frac{f'(c_n)}{g'(c_n)} } { } { } { } { } { } { } }{.} 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 \mavergleichskette {\vergleichskette { \frac{f'( a )}{g'( a )} } { = }{ w } { }{ } { }{ } { }{ } }{.} Therefore, also the left-hand side converges to $w$, and, because of \mavergleichskette {\vergleichskette { f( a ) } { = }{ g( a ) } { = }{ 0 } { }{ } { }{ } }{,} this means that \mathl{\frac{f(x_n)}{g(x_n)}} converges to $w$.
\inputbeispiel {
The polynomials \mathdisp {3x^2-5x-2 \text{ and } x^3-4x^2+x+6} { } have both a zero for \mavergleichskette {\vergleichskette {x } { = }{2 } { }{ } { }{ } { }{ } }{.} It is therefore not immediately clear whether the limit \mathdisp {\operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 3x^2-5x-2}{x^3-4x^2+x+6}} { } exists. Applying twice L'Hôpital's rule, we get the existence and \mavergleichskettedisp {\vergleichskette { \operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 3x^2-5x-2}{x^3-4x^2+x+6} } { =} { \operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 6x-5}{3x^2-8x+1} } { =} { \frac{7}{-3} } { =} { - \frac{7}{3} } { } { } }{.}