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WMA12/01 Oct 2020, Question 3: Stationary Points and Turning Points

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

The question, in short

For the cubic f(x) = ax³ − x² + bx + 4 with constants a and b, use given remainder and factor information to find a and b, differentiate, and locate stationary points.

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

Part (a) · 2 marks

Given the remainder −108 when dividing by (x + 4), use the remainder theorem to demonstrate that 16a + b = 24.

  1. 1f(−4) = −108
  2. 2a(−4)³ − (−4)² + b(−4) + 4 = −108
  3. 3−64a − 16 − 4b + 4 = −108
  4. 4−64a − 4b = −96, so 16a + b = 24

AnswerShown: f(−4) = −64a − 16 − 4b + 4 = −108, so 16a + b = 24

Part (b) · 3 marks

Using also that (2x − 1) is a factor, determine the numerical values of a and b.

  1. 1f(½) = 0: a/8 − 1/4 + b/2 + 4 = 0
  2. 2Multiply by 8: a + 4b = −30
  3. 3Solve with 16a + b = 24: b = 24 − 16a, so a + 96 − 64a = −30
  4. 463a = 126, so a = 2 and b = −8

Answera = 2, b = −8

Part (c) · 1 mark

Differentiate f(x) to obtain f′(x).

  1. 1f(x) = 2x³ − x² − 8x + 4
  2. 2f′(x) = 6x² − 2x − 8

Answerf′(x) = 6x² − 2x − 8

Part (d) · 4 marks

Using your derivative, work out the exact coordinates of the stationary points on y = f(x).

  1. 16x² − 2x − 8 = 0, so 3x² − x − 4 = 0
  2. 2(3x − 4)(x + 1) = 0, so x = 4/3 or x = −1
  3. 3x = −1: y = −2 − 1 + 8 + 4 = 9
  4. 4x = 4/3: y = 2(64/27) − 16/9 − 32/3 + 4 = −100/27

Answer(4/3, −100/27) and (−1, 9)

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Also searched as: WMA12 October 2020 · WMA12/01 · wma12 oct20 · IAS Maths Pure Mathematics 2 October 2020 · question 3.
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