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4PM1/01 Jun 2022, Question 7: Solving Cubic Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 15 marks

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The question, in short

Given a cubic gradient function and a point (4, −104) on curve C, work with the curve's turning points and its passage through the origin.

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

Part (ai) · 4 marks

Using the given derivative and point, verify that the curve passes through (0, 0).

  1. 1f(x) = ∫ (4x³ − 12x² − 19x + 12) dx = x⁴ − 4x³ − (19/2)x² + 12x + D
  2. 2At (4, −104): −104 = 256 − 256 − 152 + 48 + D
  3. 3−104 = −104 + D ⇒ D = 0
  4. 4So f(0) = 0 and C passes through the origin

Answerf(x) = x⁴ − 4x³ − (19/2)x² + 12x with constant 0, so f(0) = 0 and C passes through the origin

Part (aii) · 3 marks

Show that the curve has a maximum where x = 0.5.

  1. 1f′(0.5) = 4(0.125) − 12(0.25) − 19(0.5) + 12 = 0.5 − 3 − 9.5 + 12 = 0
  2. 2f″(x) = 12x² − 24x − 19
  3. 3f″(0.5) = 3 − 12 − 19 = −28 < 0, so maximum

Answerf′(0.5) = 0 and f″(0.5) = −28 < 0, so a maximum at x = 0.5

Part (bi) · 5 marks

Given A's x coordinate is negative, find the coordinates of both turning points A and B.

  1. 1(2x − 1) is a factor of f′(x): f′(x) = (2x − 1)(2x² − 5x − 12)
  2. 2= (2x − 1)(2x + 3)(x − 4), so x = −3/2 or x = 4
  3. 3f(−3/2) = 81/16 + 27/2 − 171/8 − 18 = −333/16
  4. 4f(4) = −104
  5. 5A(−3/2, −333/16), B(4, −104)

AnswerA(−3/2, −333/16), B(4, −104)

Part (bii) · 3 marks

State whether each of the two turning points is a maximum or minimum.

  1. 1f″(−3/2) = 27 + 36 − 19 = 44 > 0, so A is a minimum
  2. 2f″(4) = 192 − 96 − 19 = 77 > 0, so B is a minimum

AnswerA and B are both minimum points

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