Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 9D / b / Solution

A function is differentiable at $a$ when the finite limitexists. A derivative need not be continuous. For example,is differentiable everywhere and $f^{\prime }(0)=0$, but for $x\ne 0$,which has no limit at zero.

The mean value theorem states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c\in (a,b)$ satisfies