Theorem: Rolle's Theorem

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.