Rolle's Theorem

Theorem
Let $f$ be a real function which is:
 * continuous on the closed interval $\closedint a b$

and:
 * differentiable on the open interval $\openint a b$.

Let $\map f a = \map f b$.

Then:
 * $\exists \xi \in \openint a b: \map {f'} \xi = 0$

Also see

 * Extended Rolle's Theorem