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0580/22 May/Jun 2025, Question 21: Stationary Points and Turning Points

Cambridge IGCSE Mathematics 0580 · Paper 22 (Extended) · Non-calculator · 9 marks

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The question, in short

For the curve y = 3x − x³ shown crossing the x-axis at A, O, B with turning points P and Q, find exact x-intercepts, the derivative, and turning point coordinates.

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

Part (a) · 3 marks

Find the exact x-coordinates of the points A and B where the curve crosses the x-axis.

  1. 13x − x³ = 0
  2. 2x(3 − x²) = 0, so x = 0 or x² = 3
  3. 3x = ±√3
  4. 4A: −√3, B: √3

AnswerA: −√3, B: √3

Part (bi) · 2 marks

Differentiate the expression 3x − x³ with respect to x.

  1. 1Multiply by the power, then take 1 off the power: axⁿ → naxⁿ⁻¹
  2. 23x → 3
  3. 3−x³ → −3x²
  4. 4dy/dx = 3 − 3x²

Answer3 − 3x²

Part (bii) · 4 marks

Use the derivative to find the coordinates of the turning points P and Q.

  1. 1dy/dx = 3 − 3x²
  2. 2Turning points: 3 − 3x² = 0, so x = 1 or x = −1
  3. 3x = −1: y = −3 + 1 = −2; x = 1: y = 3 − 1 = 2
  4. 4P is below the x-axis: P = (−1, −2) and Q = (1, 2)

AnswerP = (−1, −2) and Q = (1, 2)

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Also searched as: 0580/22/M/J/25 · 0580_s25_qp_22 · 0580 s25 22 · question 21.
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