Skip to content

Critical Point

Definition

For a function f ⁣:X→R,f\colon X\to\mathbb{R}\text{,} a critical point of ff is a point in the domain of ff where its derivative is either zero or undefined. That is, x0x_0 is a critical point if f′(x0)=0f'(x_0)=0 or f′(x0)f'(x_0) is undefined, for x0∈X.x_0\in X\text{.}

Interpretation

Critical points are the points of the domain where the derivative gives no direction of increase or decrease: either f′(x0)=0f'(x_0)=0, so the tangent is horizontal, or f′(x0)f'(x_0) does not exist. They are the candidates for local extrema at interior points of the domain.

A cubic f with horizontal tangents at x = -1 and x = 1, where f' = 0, and a function g with a corner at a point c, where g' does not exist.

The figure shows both kinds. The cubic f(x)=x3−3x+1f(x)=x^3-3x+1 has f′(x)=3x2−3f'(x)=3x^2-3, which is zero at x=−1x=-1 and x=1x=1; there the tangents are horizontal, and ff has a local maximum and a local minimum. The function gg has a corner at cc, where the slopes from the left and from the right differ, so g′(c)g'(c) does not exist.

Both kinds are collected because of the first-order condition: at an interior point where ff is differentiable, a local extremum forces f′(x0)=0f'(x_0)=0. An interior local extremum is therefore either a point with f′(x0)=0f'(x_0)=0 or a point where ff is not differentiable, and in both cases a critical point. For a continuous function on [a,b][a,b], the critical points together with the endpoints aa and bb form the complete list of candidates for the global extrema guaranteed by the extreme value theorem.

The points with f′(x0)=0f'(x_0)=0 alone are the stationary points; for a differentiable function the two notions coincide.

Two slips are common. Solving f′(x)=0f'(x)=0 alone misses the critical points where f′f' does not exist: f(x)=x23f(x)=\sqrt[3]{x^2} has f′(x)=23x3≠0f'(x)=\frac{2}{3\sqrt[3]{x}}\neq0 for all x≠0x\neq0, yet its minimum is at 00. And a critical point must belong to the domain: 1/x1/x has no derivative at 00, but 00 is not a critical point, because 1/x1/x is not defined there.

Key Results

  • Local extrema at interior points of the domain can occur only at critical points; boundary points of the domain must be checked separately.
  • Not every critical point corresponds to an extremum.

Examples

  • For f(x)=x2,f(x)=x^2\text{,} the point x=0x=0 is a critical point.
  • For f(x)=x3,f(x)=x^3\text{,} the point x=0x=0 is a critical point.
  • For f(x)=∣x∣,f(x)=|x|\text{,} the point x=0x=0 is a critical point where the derivative is undefined.

Prerequisites

Sources

Aliases: critical number Symbols: element of, real numbers, square root

Last updated: 20 October 2026 03:02