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Stationary Point

Definition

For a differentiable function f ⁣:X→R,f\colon X\to\mathbb{R}\text{,} a stationary point of ff is a point x0∈Xx_0\in X where its derivative is zero, i.e., f′(x0)=0;f'(x_0)=0\text{;} the corresponding point (x0,f(x0))(x_0,f(x_0)) on the graph of ff is sometimes also called a stationary point.

Interpretation

A stationary point is a point where the derivative is zero: the tangent to the graph is horizontal, and to first order the function is not changing there. Stationary points are the candidates for local extrema of a differentiable function at interior points.

A wave-shaped curve with horizontal tangent lines at three points, two peaks and one valley, where the derivative is 0.

The curve in the figure has three stationary points, each marked with a horizontal tangent: two peaks, which are local maxima, and a valley between them, which is a local minimum. At each of them the curve turns from rising to falling or the other way round, so f′f' changes sign there.

A zero derivative alone does not say what happens nearby: the function may have a peak, a valley or neither, as x3x^3 at 00 shows. Whether a stationary point is a local extremum has to be checked by comparing f(x)f(x) with f(x0)f(x_0) for xx near x0x_0; the second-order condition later gives a shortcut.

Every stationary point is a critical point, but critical points also include points where f′f' does not exist; for a differentiable function the two notions coincide.

A common slip is to confuse a stationary point with a zero of ff. The condition is f′(x0)=0f'(x_0)=0, not f(x0)=0f(x_0)=0: x2+1x^2+1 has a stationary point at 00 but no zeros.

Stationary points return for functions of several variables in Math III and Math IV, with the gradient equal to zero in place of f′(x0)=0f'(x_0)=0.

Key Results

  • Not every stationary point corresponds to an extremum.
  • Stationary points must be analysed further to determine behaviour.

Examples

  • The function f(x)=x2f(x)=x^2 has a stationary point at x=0.x=0\text{.}
  • The function f(x)=x3f(x)=x^3 has a stationary point that is not an extremum.

Prerequisites

Sources

Symbols: element of, real numbers

Last updated: 20 October 2026 03:02