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WMA12/01 Jan 2020, Question 10: Stationary Points and Turning Points

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 10 marks

The question, in short

Curve y = 4x³ − 9x + k/x for x > 0, with P at x = 1/2 on C and stationary; find k, classify P, and locate the other stationary point.

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

Part (a) · 4 marks

Show that the constant k equals −3/2, using that P at x = 1/2 is a stationary point on C.

  1. 1dy/dx = 12x² − 9 − k/x²
  2. 2At x = 1/2: 12(1/4) − 9 − k/(1/4) = 0
  3. 33 − 9 − 4k = 0
  4. 44k = −6, so k = −3/2

AnswerShown: 12(1/2)² − 9 − 4k = 0 ⇒ 4k = −6 ⇒ k = −3/2

Part (b) · 2 marks

Determine whether the stationary point at P is a maximum or minimum, giving a justification.

  1. 1dy/dx = 12x² − 9 + 3/(2x²)
  2. 2d²y/dx² = 24x − 3/x³
  3. 3At x = 1/2: 12 − 24 = −12
  4. 4−12 < 0, so P is a maximum

AnswerMaximum, because d²y/dx² = −12 < 0 at x = 1/2

Part (c) · 4 marks

Use algebra to find the x coordinate of the second stationary point of C.

  1. 112x² − 9 + 3/(2x²) = 0
  2. 2Multiply by 2x²: 24x⁴ − 18x² + 3 = 0
  3. 33(4x² − 1)(2x² − 1) = 0
  4. 4x² = 1/4 gives P, so x² = 1/2
  5. 5x = √2/2 (x > 0)

Answerx = √2/2

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