# Critical point

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## 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.