Critical points

The critical points of a scalar function of several variables $f:\R^n \to \R$ are the points where all partial derivatives are zero. In other words, $\mathbf p$ is a critical point if and only if $\nabla f(\mathbf p) = \mathbf 0$. For a function of one variable $g(x)$ the values of $x$ for which $\frac{dg}{dx}=0$ are sometimes called critical numbers. To find the critical points of a function we calculate the partial derivatives and set them equal to zero. The solutions to the resulting system are the critical points.

Some critical points may be maxima or minima of the function. To test if a critical point is a maximum or minimum (or neither) we can use the second derivative test.