Fermat’s theorem (stationary points)

Let $f:(a,b)\to ℝ$ 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 $f^{\prime }(x_0)=0$.

Moreover if $f$ has a local maximum at $a$ and $f$ is differentiable at $a$ (the right derivative exists)then $f^{\prime }(a)\le 0$; if $f$ has a local minimum at $a$ then $f^{\prime }(a)\ge 0$.If $f$ is differentiable in $b$ andhas a local maximum at $b$ then $f^{\prime }(b)\ge 0$ while if it has a local minimum at $b$ then $f^{\prime }(b)\le 0$.