Interior extremum theorem

In mathematics, the interior extremum theorem, also known as Fermat's theorem, is a theorem which states that at the local extrema of a differentiable function, its derivative is always zero. It belongs to the mathematical field of real analysis and is named after French mathematician Pierre de Fermat.

By using the interior extremum theorem, the potential extrema of a function ${\displaystyle f}$, with derivative ${\displaystyle f'}$, can found by solving an equation involving ${\displaystyle f'}$. The interior extremum theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.