Higher Derivatives
Taking derivatives of derivatives of differentiable functions.
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Study Higher Derivatives
Here we consider derivatives of derivatives (of derivatives ...) and discuss consequences for the search for local extrema of functions.
Definition 1. If the derivative of a function is
differentiable, we call \((f')'\) the second
derivative. The \(n\)-th
derivative of a function is inductively defined as the derivative
of the \(n-1\)-th derivative.
For the second derivative at \(x_0\),
we write \(f''(x_0)\) or \(\frac{d^2}{dx^2}f(x_0)\). The \(n\)-th derivative is denoted by \(f^{(n)}(x_0)\) or \(\frac{d^n}{dx^n}f(x_0)\).
We call a function \(f:I\to\mathbb{R}\)
\(n\)-times differentiable if
\(f^{(n)}(x)\) exists for all \(x\in I\).
Furthermore we call a function \(f:I\to\mathbb{R}\) \(n\)-times continuously differentiable
if \(f^{(n)}:I\to\mathbb{R}\) exists
and is continuous.
Theorem 2. Let \(I:=[a,b]\) and \(f:I\to\mathbb{R}\) be differentiable. Furthermore let \(x_0\in I\) such that \(f'(x_0)=0\) and \(f'\) is differentiable in \(x_0\). Then
if \(f''(x_0)>0\), then \(f\) has a local minimum in \(x_0\);
if \(f''(x_0)<0\), then \(f\) has a local maximum in \(x_0\).
Proof: We only show the case \(f''(x_0)>0\) (the opposite case is analogous). By definition, we have \[f''(x_0)=\lim_{x\to x_0}\frac{f'(x)-f'(x_0)}{x-x_0}>0.\] Since \(f'\) is continuous in \(x_0\), we have that there exists some \(\varepsilon>0\) such that for all \(x\in I\backslash\{x_0\}\) with \(|x-x_0|<\varepsilon\) holds \[\frac{f'(x)-f'(x_0)}{x-x_0}>0.\] Since \(f'(x_0)=0\), we have that \[\begin{aligned} f'(x)<0&\quad\text{for all }x\in(x_0-\varepsilon,x_0),\\ f'(x)>0&\quad\text{for all }x\in(x_0,x_0+\varepsilon). \end{aligned}\] Therefore, \(f\) is monotonically decreasing in \((x_0-\varepsilon,x_0)\) and monotonically increasing in \((x_0,x_0+\varepsilon)\). Therefore, \(f\) has a local minimum in \(x_0\).\(\Box\)
Remark 3. Note that in the case \(f''(x_0)=0\), we cannot make a decision whether \(f\) has a local extremum there. For instance, consider the three functions \(f_1(x)=x^3\), \(f_2(x)=x^4\) and \(f_3(x)=-x^4\). We have \(f_1'(0)=f_2'(0)=f_3'(0)=0\) and, furthermore, \(f_1''(0)=f_2''(0)=f_3''(0)=0\). However, \(f_1\) has no local extremum in \(0\), \(f_2\) has a local minimum in \(0\) and \(f_3\) has a local maximum in \(0\).
Example 4.
Consider the rational function \[f(x)=\frac{x(x+5)}{x-4}=x+9+\frac{36}{x-4}.\] It can be easily seen that \(f\) has a first order pole at \(x_0=4\).
The first two derivatives of \(f\) are given by \[f'(x)=\frac{x^2-8x-20}{(x-4)^2}=\frac{(x+2)(x-10)}{(x-4)^2},\qquad f''(x)=\frac{72}{(x-4)^3}.\] The zeros of \(f\) are given by \(x_1=0\) and \(x_2=-5\).
Now we determine the set of local extrema: We have that \(f'(x)=0\) is only fulfilled for \(x_3=-2\) and \(x_4=10\). In this case, we have \(f''(x_3)=-\frac13<0\) and \(f''(x_4)=\frac13>0\). As a consequence, \(f\) has a local maximum in \(x_3=-2\) and a local minimum in \(x_4=10\). We further have\(x\in(-\infty,-2)~\Rightarrow f'(x)>0\), i.e., \(f\) is strictly monotonically increasing in \((-\infty,-2)\);
\(x\in(-2,4)~\Rightarrow f'(x)<0\), i.e., \(f\) is strictly monotonically decreasing in \((-2,4)\);
\(x\in(4,10)~\Rightarrow f'(x)<0\), i.e., \(f\) is strictly monotonically decreasing in \((4,10)\);
\(x\in(10,\infty)~\Rightarrow f'(x)>0\), i.e., \(f\) is strictly monotonically increasing in \((10,\infty)\).
\(f(x)=\sin(x)\). The zeros are given by \[\{0,\pi,-\pi,2\pi,-2\pi,3\pi,-3\pi,\ldots\}=\{n\pi\;|\;n\in\mathbb{Z}\}.\] The first two derivatives are given by \(f'(x)=\cos(x)\), \(f''(x)=-\sin(x)\). The zeros of \(f'\) are given by \[\left\{\frac\pi2,-\frac\pi2,\frac{3\pi}2,-\frac{3\pi}2,\frac{5\pi}2,\frac{5\pi}2,\ldots\right\}=\left\{\frac{2n+1}2\pi\;|\;n\in\mathbb{Z}\right\}.\] For \(x_n=\frac{2n+1}2\pi\), we have \(f''(x_n)=-\sin(\frac{2n+1}2\pi)=(-1)^{n+1}\). As a consequence, \(\sin\) has a local maximum in \(x_n=\frac{2n+1}2\pi\) if \(n\) is even and a local minimum in \(x_n=\frac{2n+1}2\pi\) if \(n\) is odd.
Definition 5 (Convexity/Concavity). A function \(f:[a,b]\rightarrow \mathbb{R}\), \(a,b\in\mathbb{R}\), \(a<b\), is called convex, if for all \(x_1<x<x_2\) in \([a,b]\) holds \[\label{eq:convex} f(x) \leq \frac{f(x_2)-f(x_1)}{x_2-x_1}\cdot(x-x_1)+f(x_1)\ .\] It is called concave, if for all \(x_1<x<x_2\) in \([a,b]\) holds \[\label{eq:concave} f(x)\geq \frac{f(x_2)-f(x_1)}{x_2-x_1}\cdot(x-x_1)+f(x_1)\ .\] If the inequalities in Definition 5 are strict then \(f\) is called strictly convex/concave.
Geometrically this means that the graph of a convex (concave) function \(f:[a,b]\rightarrow \mathbb{R}\) restricted to any subinterval \([x_1,x_2]\) of \([a,b]\) lies below (above) the secant \[s(x):=\frac{f(x_2)-f(x_1)}{x_2-x_1}\cdot(x-x_1)+f(x_1) \ .\]
Theorem 6. Let \(f:[a,b]\rightarrow \mathbb{R}\), \(a,b\in\mathbb{R}\), \(a<b\), be 2-times differentiable.
If \(f''(x)\geq 0\) for all \(x\in~(a,b)\) , then \(f\) is convex.
If \(f''(x)>0\) for all \(x\in~(a,b)\) , then \(f\) is strictly convex.
If \(f''(x)\leq 0\) for all \(x\in~(a,b)\) , then \(f\) is concave.
If \(f''(x)<0\) for all \(x\in~(a,b)\) , then \(f\) is strictly concave.
Proof: We only prove a). The other results follow analogously. Let \(x_1<x<x_2\) in \([a,b]\). Since \(f''\geq 0\) we know that \(f'\) is monotonically increasing. By the intermediate value theorem there are \(\xi_1\in~(x_1,x)\) and \(\xi_2\in~(x,x_2)\) such that \[\frac{f(x)-f(x_1)}{x-x_1} = f'(\xi_1) \leq f'(\xi_2) = \frac{f(x_2)-f(x)}{x_2-x} \ .\]
This implies \[(f(x)-f(x_1))(x_2-x) \leq (f(x_2)-f(x))(x-x_1)\] \[\Leftrightarrow f(x)(x_2-x_1) \leq (f(x_2)-f(x_1))(x-x_1)+f(x_1)(x_2-x_1)\] \[\Leftrightarrow f(x) \leq \frac{f(x_2)-f(x_1)}{x_2-x_1}(x-x_1)+f(x_1)\ .\] \(\Box\)
Definition 7. Inflection point Let \(f:[a,b]\rightarrow \mathbb{R}\) be a function. We say that \(x_0\in ~(a,b)\) is an inflection point of \(f\) if there is an \(\varepsilon>0\) with \([x_0-\varepsilon,x_0+\varepsilon]\subset[a,b]\) such that one of the following two statements holds true:
\(f\) is convex on \([x_0-\varepsilon,x_0]\) and concave on \([x_0,x_0+\varepsilon]\).
\(f\) is concave on \([x_0-\varepsilon,x_0]\) and convex on \([x_0,x_0+\varepsilon]\).
Theorem 8. Let \(f:[a,b]\rightarrow \mathbb{R}\) be 3-times continuously differentiable and \(x_0\in~(a,b)\).
If \(x_0\) is an inflection point, then \(f''(x_0)=0\).
If \(f''(x_0)=0\) and \(f'''(x_0)\neq 0\), then \(x_0\) is an inflection point.
Proof: a) follows from Theorem 6 and continuity of \(f''\).
b) If \(f''(x_0)=0\) and \(f'''(x_0)\neq 0\), then by continuity of \(f'''\) there is an \(\varepsilon>0\) with \([x_0-\varepsilon,x_0+\varepsilon]\subset[a,b]\) such that \(f'''\) does not have a zero on \([x_0-\varepsilon,x_0+\varepsilon]\). This implies that \(f''\) is strictly monotonic on \([x_0-\varepsilon,x_0+\varepsilon]\). Thus either
\(f''>0\) on \([x_0-\varepsilon,x_0]\) and \(f''< 0\) on \([x_0,x_0+\varepsilon]\) or
\(f''< 0\) on \([x_0-\varepsilon,x_0]\) and \(f''> 0\) on \([x_0,x_0+\varepsilon]\)
holds true. In case i), \(f\) is
convex on \([x_0-\varepsilon,x_0]\) and
concave on \([x_0,x_0+\varepsilon]\)
and in case ii) the reversed behavior is given. Therefore \(x_0\) is an inflection point. \(\Box\)
The aim of a so-called curve discussion of a function \(f:D\rightarrow \mathbb{R}\), \(D\subset \mathbb{R}\), is to determine its
qualitative and quantitative behaviour. We give a short list of things
that have to be investigated/determined:
Domain of definition \(D\)
Symmetries
\(f\) is symmetrical with respect to the \(y\)-axis if \(f(x)=f(-x)\) for all \(x\) in the domain of definition. In this case \(f\) is called an even function.
\(f\) is point-symmetrical with respect to the origin if \(f(-x)=-f(x)\) for all \(x\) in the domain of definition. In this case \(f\) is called an odd function.
Poles
Behaviour for \(x\longrightarrow \pm\infty\), asymptotes
A straight line \(g(x)=ax+b\), \(a,b\in\mathbb{R}\), is called an asymptote of \(f\) for \(x\rightarrow\pm\infty\) if \(\lim_{x\rightarrow\pm\infty}(f(x)-(ax+b)) = 0\). In this case the coefficients \(a,b\) can be successively determined by \[a = \lim_{x\rightarrow\pm\infty}\frac{f(x)}{x}\ ,\] \[b = \lim_{x\rightarrow\pm\infty}(f(x)-ax) \ .\]
Zeros
Extrema, monotonicity behaviour
Inflection points, convexity/concavity behaviour
Function graph
Example 9. We want to give a complete curve discussion for the rational function \[f(x)=\frac{2x^2+3x-4}{x^2} \ .\]
Domain of definition: \(D=\mathbb{R}\backslash\{0\}\)
Symmetries: \(f\) is neither an even nor an odd function.
Poles: \(x_0=0\) is a pole of order \(2\), \(\lim_{x\nearrow 0}f(x)=-\infty=\lim_{x\searrow 0}f(x)\)
Behaviour for \(x\longrightarrow \pm\infty\), asymptotes: \(\lim_{x\rightarrow\pm\infty}\frac{f(x)}{x} = 0\), \(\lim_{x\rightarrow\pm\infty} f(x) =2\). Thus the horizontal line at \(y=2\) is an asymptote of \(f\) for \(x\longrightarrow \infty\) and also for \(x\longrightarrow -\infty\).
Zeros: \(f(x)=0 \ \Leftrightarrow \ 2x^2+3x-4=0 \ \Leftrightarrow \ x=x_{1,2}=\frac{1}{4}(-3\pm\sqrt{41})\)
\(x_1\approx-2.35\), \(x_2\approx 0.85\)Extrema, monotonicity behaviour:
\(f'(x)=\frac{-3x+8}{x^3}=0 \ \Leftrightarrow \ x=x_3= \frac{8}{3}\), \(y_3:=f(x_3)\approx 2.56\)
\(f''(x)=\frac{6x-24}{x^4}\)
\(f''(x_3)<0 \ \Rightarrow \ \) \(f\) has a local maximum at \(x_3\).
\[f'(x) \quad \left\{ \begin{array}{lll} <0 &, \frac{8}{3} <x <\infty & \text{, strictly }\ \text{monotonically decreasing} \\ >0 &, 0<x<\frac{8}{3} & \text{, strictly }\ \text{monotonically increasing} \\ <0 &, -\infty<x<0 & \text{, strictly }\ \text{monotonically decreasing} \end{array} \right.\]
Inflection points, convexity/concavity behaviour:
\(f''(x)=0 \ \Leftrightarrow \ x=x_4=4\), \(y_4=f(x_4)=\frac{5}{2}\)
\(f'''=\frac{96-18x}{x^5}\)
\(f'''(x_4)>0 \ \Rightarrow \ \) \(x_4\) is an inflection point.
\[f''(x) \quad \left\{ \begin{array}{lll} >0 &, 4 <x <\infty &, \text{ strictly}\ \text{ convex} \\ <0 &, 0<x<4 &, \text{ strictly}\ \text{ concave} \\ <0 &, -\infty<x<0 &, \text{ strictly}\ \text{concave} \end{array} \right.\]
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