Stationary point

Definition

A stationary point occurs where f′(x) = 0. It may be a local maximum, a local minimum, or a point of inflection, and is classified by testing the sign of the derivative on either side or using the second derivative.

Worked example

For f(x) = x³ − 3x, f′(x) = 3x² − 3 = 0 gives x = ±1: a local maximum at x = −1 and a local minimum at x = 1.

Where it appears in the exam

Paper 1 asks for the turning points of a cubic function before you sketch it, or for the coordinates of a local maximum or minimum.

Step-by-step method

1. Differentiate the function.
2. Set f′(x) = 0 and solve for x.
3. Substitute each x-value into f(x) to find the y-coordinate.
4. Classify each point using f″(x) or the sign of f′(x) on either side.

Common mistake

Substituting the x-value into f′(x) instead of f(x) to find the y-coordinate.

Related calculus terms

• First principles
• Point of inflection
• Optimisation

Frequently asked questions

What does stationary point mean in Grade 12 Maths?

A stationary point occurs where f′(x) = 0. It may be a local maximum, a local minimum, or a point of inflection, and is classified by testing the sign of the derivative on either side or using the second derivative.

Which NSC paper tests stationary point?

Stationary point belongs to the Calculus section of CAPS Grade 12 Mathematics, which is examined in Paper 1.

How do I practise questions involving stationary point?

Snap a question that uses stationary point into the Snap&Learn AI solver, or work through the calculus topic guide and NSC past papers.