Fermat's theorem (stationary points)

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In real analysis, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is named after a French mathematician Pierre de Fermat.

By using Fermat's theorem, the potential extrema of a function ${\displaystyle \displaystyle f}$, with derivative ${\displaystyle \displaystyle f'}$, are found by solving an equation in ${\displaystyle \displaystyle f'}$. Fermat's 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.