Let $f:\mathbb{R}\to\mathbb{R}$ be a function which is continuous on the interval $[a,b]$ and differentiable on $(a,b)$. Then there exists a number $c:a<c<b$ such that

The geometrical meaning of this theorem is illustrated in the picture:

The dashed line connects the points $(a,f(a))$ and $(b,f(b))$. There is $c$ between $a$ and $b$ at which the tangent to $f$ has the same slope as the dashed line.

The mean-value theorem is often used in the integral context: There is a $c\in[a,b]$ such that