Real functions Differentiation Definition

Critical point

Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ , and let $a \in I$ . The point $a$ is a critical point of $f$ if either $f$ is not differentiable at $a$ , or $f$ is differentiable at $a$ with $f'(a) = 0$ . In the second case, $a$ is also called a stationary point.

Remarks

By Fermat's theorem, each local extremum of $f$ in $I$ is a critical point. The converse fails: $x^3$ has a critical point at $0$ , but no local extremum. For $|x|$ , the point $0$ is a critical point where $f$ is not differentiable.

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Limit point of a set definition
Limit of a function definition
Derivative definition
Critical point definition

Tag	`0GEQ42PTGB`
URI	`m2://0GEQ42PTGB`
Aliases	`stationary-point`
Kind	definition
Writing	W1 stated
MSC	26A24
Defines	Critical point
Uses	Derivative

Critical point

Definition

Remarks

Reading path

Further reading

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