Editing Vector calculus (section)

=== Optimization ===
{{main|Mathematical optimization}}
For a continuously differentiable [[function of several real variables]], a point {{math|''P''}} (that is, a set of values for the input variables, which is viewed as a point in {{math|'''R'''<sup>''n''</sup>}}) is '''critical''' if all of the [[partial derivative]]s of the function are zero at {{math|''P''}}, or, equivalently, if its [[gradient]] is zero. The critical values are the values of the function at the critical points.

If the function is [[smooth function|smooth]], or, at least twice continuously differentiable, a critical point may be either a [[local maximum]], a [[local minimum]] or a [[saddle point]]. The different cases may be distinguished by considering the [[eigenvalue]]s of the [[Hessian matrix]] of second derivatives.

By [[Fermat's theorem (stationary points)|Fermat's theorem]], all local [[maxima and minima]] of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.