Let $f\colon (a,b)\to \mathbb R$ be a continuous function and suppose that $x_0\in (a,b)$ is a local extremum of $f$ If $f$ is differentiable in $x_0$ then $f'(x_0)=0$

Moreover if $f$ has a local maximum at $a$ and $f$ is differentiable at $a$ (the right derivative exists) then $f'(a)\le 0$ if $f$ has a local minimum at $a$ then $f'(a)\ge 0$ If $f$ is differentiable in $b$ and has a local maximum at $b$ then $f'(b)\ge 0$ while if it has a local minimum at $b$ then $f'(b)\le 0$