Critical point

A point where the first derivative of a scalar function vanishes

Critical point

A critical point of a differentiable function $f:U\to \mathbb{R}$ (with $U\subseteq \mathbb{R}^n$ ) is a point $a\in U$ such that

\nabla f(a)=0,

equivalently, the Fréchet derivative $Df(a)$ is the zero linear map.

Critical points are candidates for local extrema but need not be extrema. Higher-order information, such as the Hessian matrix , is used to refine classification (see second derivative tests ).

Examples:

For $f(x)=x^3$ , the point $a=0$ is a critical point, but $0$ is not a local maximum or minimum.
For $f(x,y)=x^2+y^2$ , the point $(0,0)$ is a critical point and is a (global) minimum.