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.

Comments

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