Critical point for a function of multiple variables

Definition

Definition in terms of gradient vector

Suppose $f$ is a function of a vector variable ${\overline {x}}$ with coordinates $x_{1},x_{2},\dots ,x_{n}$. Suppose ${\overline {c}}$, with coordinates $c_{1},c_{2},\dots ,c_{n}$, is a point in the interior of the domain of $f$. We say that ${\overline {c}}$ is a critical point for $f$ if the gradient vector of $f$ at ${\overline {c}}$ is either the zero vector or does not exist.