Rolle's theorem

Throw a ball straight up: it leaves your hand and later returns to the same height. Somewhere in between it must stop rising and start falling, and at that instant its vertical velocity is zero. Rolle's theorem is this everyday fact stated for any smooth function that starts and ends at the same value.

Statement: if f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) with f'(c) = 0. Proof: by the extreme value theorem f attains a max and a min on [a, b]; if both occur at the endpoints then f is constant and f' = 0 throughout, otherwise an extremum lies in the interior and Fermat's theorem gives f'(c) = 0 there.

Each hypothesis is essential. Without differentiability inside, f(x) = |x| on [-1, 1] satisfies f(-1) = f(1) but has no point of zero derivative. Without continuity up to the endpoints the conclusion can also fail. The theorem only promises existence of at least one such c; it does not tell you where it is or how many there are.