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WMA12/01 Jun 2019, Question 5: Stationary Points, Increasing and Optimising

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

The question, in short

Watch company profit modelled by P = 12x − x^(3/2) − 120, P in thousands of pounds, x the selling price in pounds. Use calculus to find and confirm the maximum annual profit.

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

Part (a) · 6 marks

Using calculus, find the selling price x giving maximum annual profit and state the maximum profit in thousands of pounds.

  1. 1dP/dx = 12 − (3/2)x^(1/2)
  2. 2Set dP/dx = 0: (3/2)x^(1/2) = 12, so x^(1/2) = 8
  3. 3x = 64
  4. 4P = 12 × 64 − 64^(3/2) − 120 = 768 − 512 − 120 = 136
  5. 5Maximum profit = 136 thousand pounds = £136 000

Answer£136 000 (when x = 64)

Part (b) · 2 marks

Use calculus, such as the second derivative, to justify that the profit found in part a is a maximum.

  1. 1d²P/dx² = −(3/4)x^(−1/2)
  2. 2At x = 64: d²P/dx² = −(3/4) × 1/8 = −3/32
  3. 3−3/32 < 0, so the profit is a maximum

AnswerShown: d²P/dx² = −(3/4)x^(−1/2) = −3/32 < 0 at x = 64, so it is a maximum

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Also searched as: WMA12 June 2019 · WMA12/01 · wma12 jun19 · IAS Maths Pure Mathematics 2 June 2019 · question 5.
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