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.

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.