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0580/41 Oct/Nov 2023, Question 11: Stationary Points and Turning Points

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 10 marks

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The question, in short

Differentiate a cubic, locate its two stationary points from the curve equation, then classify each given stationary point as maximum or minimum with working.

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

Part (a) · 2 marks

Find the derivative of x³ − 4x² − 3x.

  1. 1Differentiate term by term: bring the power down, then reduce it by 1.
  2. 2x³ → 3x², −4x² → −8x, −3x → −3
  3. 3Answer: 3x² − 8x − 3

Answer3x² − 8x − 3

Part (b) · 5 marks

For y = x³ − 4x² − 3x, find the coordinates of both stationary points, showing working.

  1. 1Stationary points: 3x² − 8x − 3 = 0
  2. 2(3x + 1)(x − 3) = 0, so x = 3 or x = −1/3
  3. 3x = 3: y = 27 − 36 − 9 = −18
  4. 4x = −1/3: y = −1/27 − 4/9 + 1 = (−1 − 12 + 27)/27 = 14/27
  5. 5Points (3, −18) and (−1/3, 14/27)

Answer(3, −18) and (−1/3, 14/27)

Part (c) · 3 marks

Using the given stationary points (3, −18) and (−1/3, 14/27), decide which is a maximum and which a minimum, showing working.

  1. 1d²y/dx² = 6x − 8
  2. 2At x = 3: 6 × 3 − 8 = 10 > 0, so (3, −18) is a minimum
  3. 3At x = −1/3: 6 × (−1/3) − 8 = −10 < 0, so (−1/3, 14/27) is a maximum

Answer(3, −18) minimum with reason; (−1/3, 14/27) maximum with reason

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Also searched as: 0580/41/O/N/23 · 0580_w23_qp_41 · 0580 w23 41 · question 11.
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