Elementary Calculus: Rolle's Theorem

ROLLE'S THEOREM

Suppose that f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If

f(a) = f(b) = 0,

then there is at least one point c strictly between a and b where f has derivative zero; i.e.,

f'(c) = 0 for some c in (a, b).

Geometrically, the theorem says that a differentiable curve touching the x-axis at a and b must be horizontal for at least one point strictly between a and b.

PROOF

We may assume that [a, b] is the domain of f. By the Extreme Value Theorem, f has a maximum value M and a minimum value m in [a, b]. Since f(a) = 0, m ≤ 0 and M ≥ 0 (see Figure 3.8.11).

Case 1	M = 0 and m = 0. Then f is the constant function f(x) = 0, and therefore f'(c) = 0 for all points c in (a, b).
Case 2	M > 0. Let f have a maximum at c, f(c) = M. By the Critical Point Theorem, f has a critical point at c. c cannot be an endpoint because the value of f(x) is zero at the endpoints and positive at x = c. By hypothesis, f'(x) exists at x = c. It follows that c must be a critical point of the type f'(c) = 0.
Case 3	m < 0. We let f have a minimum at c. Then as in Case 2, c is in (a, b) and f'(c) = 0.