# Critical point

## Contents

## Definition

### For a function of one variable

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

Then, we say that is a **critical point** for if either the derivative equals zero *or* is not differentiable at (i.e., the derivative does not exist).

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

The value is termed the critical value.

The term *critical point* is also sometimes used for the corresponding point in the graph of .

### 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.