Rolle's theorem

If the real function f is continuous on the proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists c in (a, b) such that f′(c) = 0.

In calculus and real analysis, Rolle's theorem (or lemma) states that a real-valued differentiable function which attains equal values at two distinct points must have a stationary point somewhere between them, that is, a point where its derivative is zero. The theorem is named after Michel Rolle.