Rolle’s theorem

Rolle’s theorem is a simple but important result, familiar to anyone who has moved just beyond elementary calculus into the beginnings of analysis. Essentially it tells us that if a differentiable function has equal values at a and b, then somewhere between those two points it must have a local maximum or a local minimum.

A more formal statement of the theorem, typical of those given in modern textbooks, is as follows.

Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there exists a point c in (a, b) for which f ́(c) = 0.