Mean Value Theorem

Theorem
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

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

Proof 1

 * Mean-value-theorem.png

Also see

 * Mean Value Theorem for Integrals