Interior extremum theorem

In calculus and real analysis, the interior extremum theorem states that any local extremum of a real function at which it is differentiable is a stationary point. It is also known as Fermat's theorem, named after the French mathematician Pierre de Fermat.

The interior extremum theorem gives a necessary, but not sufficient condition for local extrema at which the function is differentiable, as some stationary points are not local extrema.

The second derivative, if non-zero, can be used to determine whether a local extremum at which the function is twice differentiable is a maximum or a minimum. However, the second derivative can be zero at local extrema.