CAL1_L23

Extrema of real valued functions of one variable

Theorem 1(Theorem).

For some integer $n\ge 1$, let $f:\mathbb{R}\to \mathbb{R}$ have continuous $n$th derivative in the open interval $(a,b)$. Suppose also that for some $c\in (a,b)$, we have

$$f^{\prime }(c)=f^{\prime \prime }(c)=\cdots =f^{(n-1)}(c)=0,\text{ but }{f}^{(n)}(c)\ne 0.$$

Then for even $n$, $f$ has a local minimum at $c$ if $f^{(n)}(c)>0$, and a local maximum at $c$ if $f^{(n)}(c)<0$. If $n$ is odd, there is neither a local maximum nor a local minimum at $c$.

Proof.

Since$f^{(n)}(c)\ne 0$ and $f^{(n)}$ is continuous at $c$, there exists an interval $B(c)$ such that for every $x$ in $B(c)$, $f^{(n)}(x)$ will have the same sign as $f^{(n)}(c)$. Now by Taylor’s formula for functions from $\mathbb{R}$ to $\mathbb{R}$, for every $x\in B(c)$,

$$\begin{array}{r}f(x)-f(c)=\frac{f^n(x_1)}{n!}(x-c{)}^n,\end{array}$$

where $x_1\in B(c)$. If $n$ is even, this equation implies $f(x)\ge f(c)$ when $f^{(n)}(c)>0$ and $f(x)\le f(c)$ when $f^{(n)}(c)<0$. If $n$ is odd, it is clear that there can be no extremum at $c$.□

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Extrema of real valued functions of several variables

Given the existence of each partial derivative $D_{k}f(\mathbf{a})$ at some interior point $\mathbf{a}$ in the domain of $f$, every partial derivative being zero is a necessary condition for $f$ to have a local maximum or a local minimum at $\mathbf{a}$. We can also state this in terms of directional derivatives by saying that $f^{\prime }(\mathbf{a};\mathbf{u})$ must be zero for every direction $\mathbf{u}$. This is not a sufficient condition, however.

Definition 2(Definition).

If $f$ is differentiable at $\mathbf{a}$ and if $\nabla f(\mathbf{a})=0$, the point $\mathbf{a}$ is called a stationary point of $\mathbf{f}$. A stationary point is called a saddle point if every $n$-ball $B(\mathbf{a})$ contains points $\mathbf{x}$ such that $f(\mathbf{x})>f(\mathbf{a})$ and other points such that $f(\mathbf{x})<f(\mathbf{a})$.

To determine whether a function of $n$ variables has a local maximum, a local minimum, or a saddle point at a stationary point $\mathbf{a}$, we must determine the algebraic sign of $f(\mathbf{x})-f(\mathbf{a})$ for all $\mathbf{x}$ in a neighborhood of $\mathbf{a}$. As in the one-dimensional case, this is done with the help of Taylor’s formula. If the partial derivatives of $f$ are differentiable on an $n$-ball $B(\mathbf{a})$ then

$$f(\mathbf{a}+\mathbf{t})-f(\mathbf{a})=\nabla f(\mathbf{a})\cdot \mathbf{t}+\frac{1}{2}{f}^{\prime \prime }(\mathbf{z};\mathbf{t}),$$

where $\mathbf{z}$ lies on the line segment $L(\mathbf{a},\mathbf{a}+\mathbf{t})$. At a stationary point we have $\nabla f(\mathbf{a})=0$, so

$$f(\mathbf{a}+\mathbf{t})-f(\mathbf{a})=\frac{1}{2}{f}^{\prime \prime }(\mathbf{z};t).$$

Therefore, as $\mathbf{a}+\mathbf{t}$ ranges over $B(\mathbf{a})$, the algebraic sign of $f(\mathbf{a}+\mathbf{t})-f(\mathbf{a})$ is determined by that of $f^{\prime \prime }(\mathbf{z};\mathbf{t})$. We would like to relate the sign of $f^{\prime \prime }(\mathbf{z};\mathbf{t})$ to that of $f^{\prime \prime }(\mathbf{a};\mathbf{t})$. Define

$$\Vert \mathbf{t}{\Vert }^2|E(\mathbf{t})|=\frac{1}{2}{f}^{\prime \prime }(\mathbf{z};\mathbf{t})-\frac{1}{2}{f}^{\prime \prime }(\mathbf{a};\mathbf{t})$$

and note that

$$\Vert \mathbf{t}{\Vert }^2|E(\mathbf{t})|\le \frac{1}{2}\sum _{i=1}^n\sum _{j=1}^n|D_{i,j}f(\mathbf{z})-D_{i,j}f(\mathbf{a})|\Vert \mathbf{t}{\Vert }^2.$$

This shows us that $E(\mathbf{t})\to 0$ as $\mathbf{t}\to 0$ if the second order partial derivatives of $f$ are continuous at $\mathbf{a}$. Now, we can write

$$f(\mathbf{a}+\mathbf{t})-f(\mathbf{a})=\frac{1}{2}{f}^{\prime \prime }(\mathbf{a};\mathbf{t})+\Vert \mathbf{t}{\Vert }^{2}E(\mathbf{t}).$$

We will show that this equation allows us to say that the sign of $f(\mathbf{a}+\mathbf{t})-f(\mathbf{a})$ is the same as that of $\frac{1}{2}{f}^{\prime \prime }(\mathbf{a};\mathbf{t})$ in a neighborhood of $\mathbf{a}$.

Theorem 3(Theorem).

Assume that the second order partial derivatives $D_{i,j}f$ exist in an $n$-ball $B(\mathbf{a})$ and are continuous at $\mathbf{a}$, where $\mathbf{a}$ is a stationary point of $f$. Let

$$Q(\mathbf{t})=\frac{1}{2}{f}^{\prime \prime }(\mathbf{a};\mathbf{t})=\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^n{D}_{i,j}f(\mathbf{a})t_i{t}_j.$$

1. If $Q(\mathbf{t})>0$ for all $\mathbf{t}\ne 0$, $f$ has a relative minimum at $\mathbf{a}$.
2. If $Q(\mathbf{t})<0$ for all $\mathbf{t}\ne 0$, $f$ has a relative maximum at $\mathbf{a}$.
3. If $Q(t)$ takes both positive and negative values, then $f$ has a saddle point at $\mathbf{a}$.

Proof
The function $Q$ is continuous at each point $\mathbf{t}$ in ${\mathbb{R}}^n$. Let $S=\{\mathbf{t}|\Vert \mathbf{t}\Vert =1\}$. If $Q(\mathbf{t})>0$ for all $\mathbf{t}\ne 0$, then $Q(\mathbf{t})$ is positive on $S$. Since $S$ is compact, $Q$ has a minimum on $S$ (call it $m$), and $m>0$. Now, $Q(c\mathbf{t})=c^{2}Q(\mathbf{t})$ for all real $\mathbf{c}$. Taking $c=1/\Vert \mathbf{t}\Vert$ when $\mathbf{t}\ne 0$, we see that $c\mathbf{t}\in S$, and $Q(c\mathbf{t})\ge m$, so $Q(\mathbf{t})\ge \Vert \mathbf{t}{\Vert }^{2}m$. So,

$$f(\mathbf{a}+\mathbf{t})-f(\mathbf{a})=Q(\mathbf{t})+\Vert \mathbf{t}{\Vert }^{2}E(\mathbf{t})\ge m\Vert \mathbf{t}{\Vert }^2+\Vert \mathbf{t}{\Vert }^{2}E(\mathbf{t}).$$

Since $E(\mathbf{t})\to 0$ as $\mathbf{t}\to 0$, there is $r>0$ such that $|E(\mathbf{t})|<m/2$ whenever $0<\Vert \mathbf{t}\Vert <r$. For such $\mathbf{t}$ we have

$$f(\mathbf{a}+\mathbf{t})-f(\mathbf{a})>m\Vert \mathbf{t}{\Vert }^2-\frac{1}{2}m\Vert \mathbf{t}{\Vert }^2>0.$$

Therefore $f$ has a relative minimum at $\mathbf{a}$.

[!Corollary]
Let $f$ be a real-valued function with continuous second-order partial derivatives exist in a ball $B(\mathbf{a})$ and are continuous at a stationary point $\mathbf{a}\in {\mathbb{R}}^2$. Let

$$A=D_{1,1}f(\mathbf{a})B=D_{1,2}f(\mathbf{a})C=D_{2,2}f(\mathbf{a}).$$

Let

$$\Delta =\det \left[\begin{array}{cc}A& B\\ B& C\end{array}\right]=AC-B^2.$$

Then, we have

1. If $\Delta >0$ and $A>0$, $f$ has a relative minimum at $\mathbf{a}$.
2. If $\Delta >0$ and $A<0$, $f$ has a relative maximum at $\mathbf{a}$.
3. If $\Delta <0$, $f$ has a saddle point at $\mathbf{a}$.