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.

Facts