The idea of critical points can be generalized to multivariable functions. For a function of several variables, a critical point occurs where all the elements of the gradient vector are zero or undefined.

Critical points are important in multivariable calculus because they can indicate locations of local maxima, local minima, or saddle points of the function. The specific nature of each critical point can be determined using the second partial derivative test.

Example

Consider the function . To find its critical point, we first compute the gradient:

Next, we set each component of the gradient to zero:

Solving these equations, we find that the only critical point is at .