# Derivatives

### Topics

Rolle's Theorem says:

Let *f* be a function that

- is continuous on the closed interval [
*a*,*b*]

- is differentiable on the open interval (
*a*,*b*), and

- has
*f*(*a*) =*f*(*b*).

Then there is some *c* in the open interval (*a*, *b*) with *f*'(*c*) = 0.

Sometimes the third condition is stated as *f *(*a*) = *f *(*b*) = 0, but for the proof, it doesn't matter.

In pictures, we're saying suppose *f* is a nice smooth function with the same starting and ending height:

If *f* increases or decreases from its starting height, it needs to turn around and come back in order to end at the same height it started at:

Since *f* is a nice smooth differentiable function, its derivative at that turn-around point must be 0:

If *f* doesn't go up or down from its starting point, then *f* is constant:

In this case, *f *' (*c*) is 0 for every value of *c* in the interval (*a*, *b*).

Rolle's Theorem is reminiscent of the Intermediate Value Theorem. Rolle's Theorem says if *f* satisfies some assumptions (more mathematically known as **hypotheses**) then *f '* will be zero at some point in (*a*, *b*). We could have a constant function, in which case *f '* will be 0 infinitely many times:

We could have a function that turns around once:

Or we could have a function that turns around many times:

Rolle's Theorem doesn't tell us where or how many times *f'* will be zero; it tells us *f '* must be zero at least once if the hypotheses are all satisfied.

### Sample Problem

Suppose *f* is not continuous on [*a*, *b*]. Then there doesn't need to be any *c* in (*a*, *b*) with *f *' (c) = 0. Here's an example:

This function is not continuous. At the point of discontinuity, *f '* doesn't exist. At all other points in the interval, *f '* is positive:

There is no point *c* in (a, b) where *f *' (c) = 0.

We found earlier that the derivative of the absolute value function doesn't exist at 0. When *x* is negative the slope of the absolute value function is *-1*; when *x* is positive the slope of the absolute value function is *-1*:

There is no value of *c* anywhere, in any interval (a, b), with *f *' (c) = 0. The derivative of the absolute value function isn't 0 anywhere.

This function doesn't have a derivative of 0 anywhere between *a* and *b*.

If a function fails any of the hypotheses, we aren't allowed to use Rolle's Theorem.