Rolle's Theorem
If f is continuous on [a, b] and differentiable on (a, b),
and f(a) = f(b) = 0, then there exists a number c in (a, b) such
that f'(c) = 0.
Context

### Proof:

**Illustrating
Rolle'e theorem**

If f is constantly equal to zero, there is nothing to prove. Hence,
assume f is not constantly equal to zero. Since f is a continuous
function on a compact set it assumes its maximum and minimum on
that set. One of them must be non-zero, otherwise the function
would be identically equal to zero. Assume for now that *f(c) #
0* is a maximum. Since *f(a) = f(b) = 0* we know that
*c* is in *(a, b)*, and therefore *f* is
differentiable at *c*. Note that
*f(x) f(c)*
since *f(c)* is a maximum.

- if x < c then
0 for all x <
c
- if x > c then
0 for all x >
c

The first inequality implies that as x approaches c from the left,
the limit must be greater than or equal to zero. The second one
says that as x approaches c from the right, the limit must be
less than or equal to zero. But since f is differentiable at c
we know that both right and left handed limits exist and must
agree. Therefore, f'(c) = 0.

The proof is similar if f(c) is a minimum. Can you see what would
change, if anything ?

**Note:** As a consequence of this proof we have shown that
if a differentiable function has a maximum or minimum in the interior
of its domain then the derivative at that point must be zero.