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0580/42 May/Jun 2022, Question 12: Gradient of a Curve at a Point

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 11 marks

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The question, in short

Curve y = x³ − kx² + 1 has gradient 6 at x = 2. Show k = 1.5, find stationary points of y = x³ − 1.5x² + 1, and sketch the curve.

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Part (a) · 5 marks

Differentiate, substitute x = 2 and gradient 6, then solve to show k = 1.5.

  1. 1dy/dx = 3x² − 2kx
  2. 2At x = 2: 3 × 2² − 2k × 2 = 6
  3. 312 − 4k = 6
  4. 44k = 6, k = 1.5

AnswerCorrect working leading to 1.5

Part (b) · 4 marks

Set the derivative of y = x³ − 1.5x² + 1 to zero and find both stationary points' coordinates, showing working.

  1. 1dy/dx = 3x² − 3x = 0
  2. 23x(x − 1) = 0, so x = 0 or x = 1
  3. 3x = 0: y = 1; x = 1: y = 1 − 1.5 + 1 = 0.5
  4. 4(0, 1) and (1, 0.5)

Answer(0, 1) and (1, 0.5)

Part (c) · 2 marks

Sketch y = x³ − 1.5x² + 1 on the axes, showing key features.

  1. 1Positive cubic shape: rises, turns, falls, turns, rises
  2. 2Maximum at (0, 1) on the positive y-axis
  3. 3Minimum at (1, 0.5) in the first quadrant; crosses the x-axis once, at a negative x

Answercorrect sketch

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