A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2014 · Q66 Marks

The equation of a curve is y = x³ + ax² + bx, where a and b are constants.

(i)In the case where the curve has no stationary point, show that a² < 3b.[3]
(ii)In the case where a = −6 and b = 9, find the set of values of x for which y is a decreasing function of x.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.

[1] dy/dx = 3x² + 2ax + b. co
[1] For no stationary point, the discriminant of dy/dx is less than zero: b² - 4ac = (2a)² - 4(3)(b) < 0. Use of discriminant on their quadratic dy/dx or other valid method
[1] → 4a² - 12b < 0 → a² < 3b. co – answer given
[1] For a = -6 and b = 9, dy/dx = 3x² - 12x + 9. Attempt at differentiation
[1] Set dy/dx = 0 to find critical points: 3x² - 12x + 9 = 0, which yields x = 1 and x = 3. co
[1] For y to be a decreasing function, dy/dx < 0, so 1 < x < 3. condone ≤

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Topic: Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source: A-Level Mathematics, Oct/Nov 2014 Paper 1 Variant 2, question 6.

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