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0580/23 May/Jun 2025, Question 21: Stationary Points and Turning Points

Cambridge IGCSE Mathematics 0580 · Paper 23 (Extended) · Non-calculator · 6 marks

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The question, in short

For the curve y = x³ − 3x² + 1, obtain its derivative and locate the stationary points, giving both coordinates.

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

Part (a) · 2 marks

Work out the derivative of x³ − 3x² + 1 with respect to x.

  1. 1Multiply by the power, then take 1 off the power: axⁿ → naxⁿ⁻¹
  2. 2x³ → 3x²
  3. 3−3x² → 2 × (−3)x = −6x
  4. 41 → 0
  5. 5dy/dx = 3x² − 6x

Answer3x² − 6x or 3x(x − 2)

Part (b) · 4 marks

Solve for the x-values where the derivative is zero, then find the matching y-values of the turning points.

  1. 1Turning points: 3x² − 6x = 0
  2. 23x(x − 2) = 0, so x = 0 or x = 2
  3. 3x = 0: y = 1
  4. 4x = 2: y = 8 − 12 + 1 = −3
  5. 5(0, 1) and (2, −3)

Answer(0, 1) and (2, −3)

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Also searched as: 0580/23/M/J/25 · 0580_s25_qp_23 · 0580 s25 23 · question 21.
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