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0580/21 May/Jun 2026, Question 24: Stationary Points and Turning Points

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Non-calculator · 9 marks

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The question, in short

Locate where the curve y = x³ − 6x² + 7 stops rising or falling, then use a sketch to classify each such point as highest or lowest.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 6 marks

Work out the coordinates of the stationary points on y = x³ − 6x² + 7.

  1. 1dy/dx = 3x² − 12x
  2. 23x² − 12x = 0
  3. 33x(x − 4) = 0, so x = 0 or x = 4
  4. 4x = 0: y = 7
  5. 5x = 4: y = 64 − 96 + 7 = −25

Answer(0, 7) and (4, −25)

Part (b) · 3 marks

Given the points (0, 7) and (4, −25), sketch the curve to decide which is a maximum and which a minimum.

  1. 1Positive x³ coefficient: curve rises, turns, falls, turns, rises
  2. 2The left turning point (0, 7) is higher, so it is a maximum
  3. 3The right turning point (4, −25) is lower, so it is a minimum

AnswerCorrect sketch of positive cubic: (0, 7) maximum, (4, −25) minimum

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Also searched as: 0580/21/M/J/26 · 0580_s26_qp_21 · 0580 s26 21 · question 24.
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