Fermat theorem

A necessary condition for a local extremum of a real-valued function. Suppose that a real-valued function $f$ is defined in a neighbourhood of a point $x_0\in\mathbf R$ and is differentiable at that point. If $f$ has a local extremum at $x_0$, then its derivative at $x_0$ is equal to zero: $f'(x_0)=0$. Geometrically this means that the tangent to the graph of $f$ at the point $(x_0,f(x_0))$ is horizontal. A condition equivalent to this for extrema of polynomials was first obtained by P. Fermat in 1629, but it was not published until 1679.

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For Fermat's theorems in number theory see Fermat great theorem; Fermat little theorem.