A-level · Beta

Stationary points and turning points

Explore maxima, minima and a stationary point that does not turn.

Move k through zero. Use the trace slider to move a tangent along the curve.

f(x)=x3−3(1)x,f′(x)=3x2−3(1)f(x)=x^3-3(1)x,\quad f\prime(x)=3x^2-3(1)
Move k through zero. Use the trace slider to move a tangent along the curve.-2.2-20-1.1-10001.1102.220xyT

The derivative changes sign at two stationary points, producing a maximum and a minimum.

Tangent gradient
-3
Stationary points
Local maximum at (-1, 2); Local minimum at (1, -2)

Blue is f(x), green is the tangent; the optional red dashed graph is f′(x). Displayed coordinates are rounded.

Predict, test and explain

Step 1 of 3

Predict

Does every stationary point have to be a maximum or minimum?

InvestigateSet k = 0. Does a horizontal tangent always mean a turning point?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Does every stationary point have to be a maximum or minimum?

f(x)=x3,f′(0)=0f(x)=x^3,\quad f\prime(0)=0

Hint

A stationary point has derivative zero.

Reveal answer

For y = x³, the tangent at zero is horizontal but the curve keeps increasing.

stationary inflection\text{stationary inflection}

Build an explanation

Hint 1

A stationary point has derivative zero.

Hint 2

For y = x³, the tangent at zero is horizontal but the curve keeps increasing.

Hint 3

Compare the result with the model, then explain why it occurs.

Worked example

This example uses fixed values, separate from the diagram controls.

Find and classify the stationary points of f(x) = x³ − 3x.

  1. Solve 3x² − 3 = 0 to get x = −1 and x = 1.
  2. The derivative changes from positive to negative at −1: a maximum.
  3. It changes from negative to positive at 1: a minimum.

Answer: Maximum (−1, 2); minimum (1, −2).

Connect this idea

Watch out: For y = x³, the tangent at zero is horizontal but the curve keeps increasing.

Specification and learning route

AQA G3 · Edexcel Pure 7.3

AS core; A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Differentiation; Solving equations; Sign of a function; Second derivatives.

AQA specification · Edexcel specification