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WMA12/01 Oct 2019, Question 10: Differentiating Polynomials

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

Curve C is y = ax³ − 3x² + 3x + b with constants a and b. Given point (2, 5) lies on C and gradient there is 7, find a and b, then show C has no turning points.

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

Part (a) · 4 marks

Use that (2, 5) satisfies the curve equation and that the derivative at x = 2 equals 7 to find the values of a and b.

  1. 1(2, 5) on C: 5 = 8a − 12 + 6 + b, so 8a + b = 11
  2. 2dy/dx = 3ax² − 6x + 3
  3. 3At x = 2: 12a − 12 + 3 = 7, so 12a = 16
  4. 4a = 4/3
  5. 5b = 11 − 32/3 = 1/3

Answera = 4/3, b = 1/3

Part (b) · 3 marks

Show that the derivative of C cannot equal zero for any real x, proving there are no turning points.

  1. 1Turning points need dy/dx = 0: 4x² − 6x + 3 = 0
  2. 2Discriminant b² − 4ac = (−6)² − 4(4)(3) = 36 − 48 = −12
  3. 3−12 < 0, so there are no real roots
  4. 4dy/dx is never zero, so C has no turning points

AnswerShown: dy/dx = 4x² − 6x + 3 has discriminant 36 − 48 = −12 < 0, so dy/dx is never 0 and C has no turning points

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Also searched as: WMA12 October 2019 · WMA12/01 · wma12 oct19 · IAS Maths Pure Mathematics 2 October 2019 · question 10.
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