Mean Value Theorem

The mean value theorem (MVT) says that, for a given arc connecting two points of a function, there is at least one point at which the slope of the tangent line is equal to the slope of the arc.

Theorem (Rolle's theorem)

Let $f(x)$ be a continuous function defined on a closed interval $[a,b]$. Suppose $f(x)$ is differentiable on the open interval $(a,b)$ and $f(a)=f(b)$. Then there exists a $c\in (a,b)$ such that $f^{\prime }(c)=0$.

Proof. If the (global) maximum and minimum are both at $x=a$ and $x=b$, then $f(x)$ is constant since $f(a)=f(b)$ (= maximum = minimum). In this case, $f^{\prime }(x)=0$ at all $x\in (a,b)$. Otherwise, $f(x)$ has a maximum or minimum value at some $c\in (a,b)$ so that $f^{\prime }(c)=0$. ■

Theorem (Mean value theorem)

Let $f(x)$ be a continuous function on $[a,b]$ such that it is differentiable on $(a,b)$. Then there exists a $c\in (a,b)$ such that

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}.$$

Proof. Let us define

$$g(x)=f(x)-\frac{f(b)-f(a)}{b-a}\cdot (x-a).$$

Then, $g(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and $g(a)=g(b)=f(a)$. Hence, by Rolle's theorem, there exists a $c\in (a,b)$ such that $g^{\prime }(c)=0$. But

$$g^{\prime }(c)=f^{\prime }(c)-\frac{f(b)-f(a)}{b-a}=0$$

so

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}.$$

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Let's see some applications of the mean value theorem.

Corollary

Suppose that the function $f(x)$ is differentiable on the open interval $I=(a,b)$ and identically $f^{\prime }(x)=0$ on $I$. Then $f(x)$ is a constant function on $I$. That is, $f(x)=C$ for all $x\in I$ where $C$ is a constant.

Proof. Let us pick an arbitrary $c\in (a,b)$ and let $C=f(c)$. We show that for all $x\in (a,b)$, $f(x)=C$. If $x=c$, then $f(x)=C$ by definition of $C$. Suppose $x\ne c$. If $x<c$, consider the closed interval $[x,c]$. Since $f(x)$ is differentiable on $(a,b)$, it is continuous on $[x,c]$. By the mean value theorem, there is a $d\in (x,c)$ such that $f(x)-f(c)=f^{\prime }(d)(x-c)$. But since $f^{\prime }(d)=0$ (by assumption), $f(x)=f(c)=C$. If $x>c$, we consider the closed interval $[c,x]$ and apply a similar argument. ■

Corollary (cor:monotone)

Suppose the function $f(x)$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$.

1. If $f^{\prime }(x)>0$ for all $x\in (a,b)$, then $f(x)$ is strictly monotone increasing on $[a,b]$.
2. If $f^{\prime }(x)<0$ for all $x\in (a,b)$, then $f(x)$ is strictly monotone decreasing on $[a,b]$.

Proof. We only show part 1. Part 2 is similar.

It suffices to show that, for any $c,d\in [a,b]$, if $c<d$ then $f(c)<f(d)$.

By assumption, $f(x)$ is continuous on $[c,d]$ and differentiable on $(c,d)$. By the mean value theorem, there exists a $t\in (c,d)$ such that $f(d)-f(c)=f^{\prime }(t)(d-c)$. But since $f^{\prime }(t)>0$, we have $f^{\prime }(t)(d-c)>0$ so that $f(d)-f(c)>0$. ■

Corollary (cor:minmax)

Suppose that the function $f(x)$ is twice differentiable and that $f^{\prime }(c)=0$ at some $c\in (a,b)$.

1. If $f^{″}(c)>0$, then $x=c$ is a local minimum point.
2. If $f^{″}(c)<0$, then $x=c$ is a local maximum point.

Proof. We prove only part 1 (Part 2 is similar).

By assumption,

$$\underset{x\to c}{lim}\frac{f^{\prime }(x)-f^{\prime }(c)}{x-c}=\underset{x\to c}{lim}\frac{f^{\prime }(x)}{x-c}=f^{″}(c)>0.$$

Let $\epsilon =f^{″}(c)(>0)$. There exists a $\delta >0$ such that, for all $x\in (a,b)$,  $0<|x-c|<\delta$ implies $|\frac{f^{\prime }(x)}{x-c}-f^{″}(c)|<\epsilon$ or

$$-\epsilon <\frac{f^{\prime }(x)}{x-c}-f^{″}(c)<\epsilon .$$

It follows that

$$\frac{f^{\prime }(x)}{x-c}>f^{″}(c)-\epsilon =0.$$

In particular, if $c-\delta <x<c$, then $x-c<0$ so $f^{\prime }(x)<0$. That is, for all $x\in (c-\delta ,c)$, $f(x)>f(c)$ (Corollary {cor:monotone}).

If $c<x<c+\delta$, we have $f^{\prime }(x)>0$ so, by the same argument (Corollary {cor:monotone}), $f(c)<f(x)$.

In summary, for all $x\in (c-\delta ,c+\delta )$, if $x\ne c$, then $f(x)>f(c)$, which shows that $f(c)$ is a local minimum value. ■

Definition (Convex, concave)

Let $f(x)$ be a function defined on an open interval $I$. $f(x)$ is said to be convex at $x=c$ if the graph of $y=f(x)$ is above its tangent line at $x=c$ in the neighbor of $x=c$, that is, if there exists a $\delta >0$ such that, for all $x\in I$,  $0<|x-c|<\delta$ implies $f(x)>f^{\prime }(c)(x-c)+f(c)$.

Similarly, $f(x)$ is said to be concave at $x=c$ if the graph of $y=f(x)$ is below its tangent line at $x=c$ in the neighbor of $x=c$, that is, there exists a $\delta >0$ such that, for all $x\in I$, $0<|x-c|<\delta$ implies $f(x)<f^{\prime }(c)(x-c)+f(c)$.

Remark. Recall that $y=f^{\prime }(c)(x-c)+f(c)$ is the equation of the tangent line of $f(x)$ at $x=c$. □

Corollary 

Let $f(x)$ be a twice differentiable function on an open interval $I$ and $c\in I$.

1. If $f^{″}(c)>0$, then $f(x)$ is convex at $x=c$. 
2. If $f^{″}(c)<0$, then $f(x)$ is concave at $x=c$.

Proof. We prove only part 1. Part 2 is similar.

Let us define $g(x)=f(x)-[f^{\prime }(c)(x-c)+f(c)]$. $g(x)$ is twice differentiable on $I$ and $g^{\prime }(x)=f^{\prime }(x)-f^{\prime }(c)$ so $g(c)=g^{\prime }(c)=0$. Also, $g^{″}(x)=f^{″}(x)$ so that $g^{″}(c)=f^{″}(c)>0$. Therefore, by Corollary {cor:minmax}, $x=c$ is a local minimum point of $g(x)$. By the definition of local minimum, it follows that $f(x)$ is convex $x=c$. ■

Example. Consider $f(x)=e^{-x^2}$. Let us find the range of $x$ where $f(x)$ is convex and where it is concave. Let us also find the extrema of $f(x)$.

$$\begin{array}{rcl}{f}^{\prime }(x)& =& -2x{e}^{-x^2},\\ f^{″}(x)& =& -2e^{-x^2}+4x^2{e}^{-x^2}=2(2x^2-1)e^{-x^2}.\end{array}$$

Solving $f^{″}(x)=0$ yields $x=\pm \frac{1}{\sqrt{2}}$. Therefore,

• $f^{″}(x)<0$ if $-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$ where $f(x)$ is concave, and
• $f^{″}(x)>0$ if $x<-\frac{1}{\sqrt{2}}$ or $x>\frac{1}{\sqrt{2}}$ where $f(x)$ is convex.

$f^{\prime }(x)=0$ only if $x=0$ and $f^{″}(0)=-2<0$. So $f(0)$ is a local maximum value (in fact, it is the global maximum value). □

As in the above example, the point at which the sign of the second-derivative changes is called an inflection point.