Critical point

Definition

For a function of one variable

Suppose $f$ is a function and $c$ is a point in the interior of the domain of $f$, i.e., $f$ is defined on an open interval containing $c$.

Then, we say that $c$ is a critical point for $f$ if either the derivative $\!f'(c)$ equals zero or $f$ is not differentiable at $c$ (i.e., the derivative $f'(c)$ does not exist).

Note that the term critical point is not used for points at the boundary of the domain.

The value $\!f(c)$ is termed the critical value.

The term critical point is also sometimes used for the corresponding point $(c,f(c))$ in the graph of $f$.

For a function of multiple variables

For further information, refer: critical point for function of multiple variables

The idea is to replace the derivative by a multiple variable notion of derivative, such as the partial derivatives, directional derivatives, or gradient vector.

Facts

• Point of local extremum implies critical point: If a function attains a local extreme value (local maximum or local minimum) at a point in the interior of its domain, then that point must be a critical point.