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WMA11/01 Jan 2022, Question 7: Differentiating Polynomials

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 11 marks

The question, in short

Use the cubic f(x) = (x + 4)(x − 2)(2x − 9) and its sketch to determine unknown constants, differentiate it, and solve a gradient inequality.

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

Part (a) · 2 marks

The curve y = f(x) − p goes through (0, 50); work out p.

  1. 1f(0) = 4 × (−2) × (−9) = 72
  2. 2f(0) − p = 50, so 72 − p = 50
  3. 3p = 22

Answerp = 22

Part (b) · 2 marks

The curve y = f(x + q) goes through the origin; state the possible q values.

  1. 1y = f(x + q) passes through (0, 0), so f(q) = 0
  2. 2q is a root of f: q = −4, q = 2 or q = 9/2

Answerq = −4, 2 or 4.5

Part (c) · 4 marks

Differentiate f(x) to obtain f′(x).

  1. 1(x + 4)(x − 2) = x² + 2x − 8
  2. 2f(x) = (x² + 2x − 8)(2x − 9) = 2x³ − 5x² − 34x + 72
  3. 3f′(x) = 6x² − 10x − 34

Answerf′(x) = 6x² − 10x − 34

Part (d) · 3 marks

Using your derivative, find all x where the gradient of y = f(x) is below −18.

  1. 16x² − 10x − 34 < −18
  2. 26x² − 10x − 16 < 0, so 3x² − 5x − 8 < 0
  3. 3(3x − 8)(x + 1) < 0, critical values x = −1 and x = 8/3
  4. 4The quadratic is negative between its roots: −1 < x < 8/3

Answer−1 < x < 8/3

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