---
tag: 0GEQ42PTGB
kind: definition
name: Critical point
aliases: [stationary-point]
msc: [26A24]
defines: 0YTCMFB4HC
uses: [0F1WR9C83K]
sources:
  - url: https://en.wikipedia.org/wiki/Critical_point_(mathematics)
    locator: Critical point of a single variable function
---

## 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|0PCVBRV9VJ]], 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.

## Explanations

### L1 human

Critical points are the candidates for local maxima and minima.
There the tangent is horizontal, or there is no tangent at all.