4PM1/01 Jun 2025, Question 9: Stationary Points and Turning Points

Part (a) · 5 marks

Using the maximum and minimum points, determine the integer values of a, b and c in the cubic.

1. 1f′(x) = 6x² + 2ax + b
2. 2f′(2) = 0: 24 + 4a + b = 0
3. 3f′(−1/3) = 0: 2/3 − 2a/3 + b = 0
4. 4Solving: a = −5, b = −4
5. 5f(2) = −9: 16 − 20 − 8 + c = −9, so c = 3

Answera = −5, b = −4, c = 3 shown

Part (bi) · 1 mark

Demonstrate that x + 1 divides the cubic.

1. 1f(−1) = −2 − 5 + 4 + 3 = 0
2. 2By the factor theorem, (x + 1) is a factor

Answerf(−1) = 0, so (x + 1) is a factor

Part (bii) · 3 marks

Fully factorise the cubic into linear factors using algebra.

1. 1Divide by (x + 1): quotient 2x² − 7x + 3
2. 22x² − 7x + 3 = (2x − 1)(x − 3)
3. 3f(x) = (x + 1)(x − 3)(2x − 1)

Answerf(x) = (x + 1)(x − 3)(2x − 1)

Part (c) · 4 marks

Find the total area of both shaded regions between the curve and the x-axis, giving an exact fraction.

1. 1M = −1, N = 1/2, P = 3
2. 2∫ f(x) dx = x⁴/2 − (5/3)x³ − 2x² + 3x
3. 3From −1 to 1/2: 117/32
4. 4From 1/2 to 3: −1375/96, area 1375/96
5. 5Total = 117/32 + 1375/96 = 863/48

Answer863/48

More past-paper questions on stationary points and turning points

• 4PM1/01 Jun 2022 Q7 · 15 marks
• 4PM1/02 Nov 2021 Q4 · 11 marks
• 4PM1/01R Jun 2020 Q4 · 6 marks
• 4PM1/02 Jun 2019 Q9 · 14 marks
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