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WMA12/01 Jun 2022, Question 8: Stationary Points, Increasing and Optimising

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Differentiate a given function involving powers of x and a reciprocal term, then use the derivative to locate the curve's stationary points.

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

Part (a) · 3 marks

Differentiate y = 256x⁴ − 304x − 35 + 27/x² with respect to x, showing each stage of working.

  1. 1Write 27/x² = 27x⁻²
  2. 2Differentiate each term: 256 × 4x³ − 304 − 2 × 27x⁻³
  3. 3dy/dx = 1024x³ − 304 − 54x⁻³

Answerdy/dx = 1024x³ − 304 − 54/x³

Part (b) · 5 marks

Using your derivative, determine the x- and y-coordinates of every stationary point on the curve.

  1. 1Set dy/dx = 0: 1024x³ − 304 − 54/x³ = 0
  2. 2Multiply by x³: 1024x⁶ − 304x³ − 54 = 0, so 512(x³)² − 152x³ − 27 = 0
  3. 3(8x³ + 1)(64x³ − 27) = 0, so x³ = −1/8 or x³ = 27/64
  4. 4x = −1/2 or x = 3/4
  5. 5x = −1/2: y = 256(1/16) + 152 − 35 + 27/(1/4) = 16 + 152 − 35 + 108 = 241
  6. 6x = 3/4: y = 256(81/256) − 228 − 35 + 27/(9/16) = 81 − 228 − 35 + 48 = −134

Answer(−1/2, 241) and (3/4, −134)

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Also searched as: WMA12 June 2022 · WMA12/01 · wma12 jun22 · IAS Maths Pure Mathematics 2 June 2022 · question 8.
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