Fermat theorem

A necessary condition for a local extremum of a real-valued function. Suppose that a real-valued function is defined in a neighbourhood of a point and is differentiable at that point. If has a local extremum at , then its derivative at is equal to zero: . Geometrically this means that the tangent to the graph of at the point 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.