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 presented as
This result can also be presented in the form:


 * $\map f {c + h} - \map f c = h \map {f'} {c + \theta h}$

for some $\theta \in \openint 0 1$.

Also known as
The mean value theorem is also known as the 'law of the mean.

Also see

 * Mean Value Theorem for Integrals