Mean value theorem

Mean value theorem: Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $c\in(a,b)$ such that

The mean value theorem is obtained from Rolle's theorem by applying Rolle to a suitable affine adjustment of $f$. It yields useful corollaries such as the mean value estimate and monotonicity criteria from the sign of the derivative .