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0580/22 Feb/Mar 2025, Question 22: 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 = x³ + x² − x with a stationary point at (1/3, −5/27), locate the remaining stationary point, classify the given one by sketching, and find k values giving fewer than 3 solutions to x³ + x² − x = k.

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

Part (a) · 5 marks

Differentiate the cubic, solve for the other x where the gradient is zero, and state both coordinates of that stationary point.

  1. 1dy/dx = 3x² + 2x − 1
  2. 2At stationary points 3x² + 2x − 1 = 0
  3. 3(3x − 1)(x + 1) = 0, so x = 1/3 or x = −1
  4. 4x = −1: y = (−1)³ + (−1)² − (−1) = −1 + 1 + 1 = 1
  5. 5(−1, 1)

Answer(−1, 1)

Part (b) · 2 marks

Draw the cubic on the supplied axes and use the shape to decide whether (1/3, −5/27) is a maximum or minimum.

  1. 1Sketch a positive cubic through the origin (y = x(x² + x − 1), roots x = 0, ≈ 0.62, ≈ −1.62)
  2. 2Maximum at (−1, 1) in the 2nd quadrant, minimum just below the x-axis at x = 1/3 in the 4th quadrant
  3. 3The curve falls then rises at x = 1/3, so (1/3, −5/27) is a minimum

AnswerCorrect sketch of positive cubic with minimum in correct quadrant; (1/3, −5/27) is a minimum

Part (c) · 2 marks

Using the sketch, work out the range of k for which the horizontal line y = k meets the curve fewer than three times.

  1. 1The line y = k meets the curve 3 times only between the turning values −5/27 and 1
  2. 2Fewer than 3 solutions when k is on or beyond a turning value
  3. 3k ≥ 1 or k ≤ −5/27

Answerk ≥ 1 or k ≤ −5/27

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Also searched as: 0580/22/F/M/25 · 0580_m25_qp_22 · 0580 m25 22 · question 22.
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