We begin with a common-sense geometrical fact:

*somewhere between two zeros of a non-constant continuous
function $f$,the function must change direction*

For a *differentiable* function, the derivative is $0$ at the point where $f$ changes direction. Thus, we expect there to be a point $c$ where the tangent is horizontal. These ideas are precisely stated by Rolle’s Theorem:

#### Rolle’s Theorem

Let $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$. If $f(a)=f(b)=0$, then there is at least one point $c$ in $(a,b)$ for which $f'(c)=0$.

Notice that both conditions on $f$ are necessary. Without either one, the statement is false!

For a *discontinuous* function, the conclusion of Rolle’s Theorem may not hold:

For a *continuous, non-differentiable* function, again this might not be the case:

Though the theorem seems logical, we cannot be sure that it is always true without a proof.

The Mean Value Theorem is a generalization of Rolle’s Theorem:

We now let $f(a)$ and $f(b)$ have values other than $0$ and look at the secant line through $(a,f(a))$ and $(b,f(b))$. We expect that somewhere between $a$ and $b$ there is a point $c$ where the tangent is parallel to this secant.

In Rolle’s Theorem, the secant was horizontal so we looked for a horizontal tangent.

That is, the slopes of these two lines are equal. This is formalized in the Mean Value Theorem.

#### Mean Value Theorem

Let $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$. Then there is at least one point $c$ in $(a,b)$ for which \[f'(c)=\frac{f(b)-f(a)}{b-a}.\]

Here, $f'(c)$ is the slope of the tangent at $c$, while $\displaystyle \frac{f(b)-f(a)}{b-a}$ is the slope of the secant through $a$ and $b$. Intuitively, we see that if we translate the secant line in the figure upwards, it will eventually just touch the curve at the single point $c$ and will be tangent at $c$. However, basing conclusions on a single example can be disastrous, so we need a proof.

#### Consequences of the Mean Value Theorem

The Mean Value Theorem is behind many of the important results in calculus. The following statements, in which we assume $f$ is differentiable on an open interval $I$, are consequences of the Mean Value Theorem:

- $f'(x)=0$ everywhere on $I$ if and only if $f$ is constant on
$I$.

- If $f'(x)=g'(x)$ for all $x$ on $I$, then $f$ and $g$ differ at
most by a constant on $I$.

- If $f'(x)>0$ for all $x$ on $I$, then $f$ is
*increasing*on $I$.

If $f'(x) < 0$ for all $x$ on $I$, then $f$ is*decreasing*on $I$.

#### Key Concepts

**Mean Value Theorem**

Let $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$. Then there is at
least one point $c$ in $(a,b)$ for which
$$f'(c) = \frac{f(b)-f(a)}{b-a}.$$