A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2017 · Q89 Marks

A curve is such that dy/dx = -x² + 5x - 4.

(i)Find the x-coordinate of each of the stationary points of the curve.[2]
(ii)Obtain an expression for d²y/dx² and hence or otherwise find the nature of each of the stationary points.[3]
(iii)Given that the curve passes through the point (6, 2), find the equation of the curve.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

dy/dx = 0 (M1) Sets dy/dx to 0 and attempts to solve leading to two values for x.
x = 1, x = 4 (A1) Both values needed
d²y/dx² = -2x + 5 (B1)
Using both of their x values in their d²y/dx² (M1) Evidence of any valid method for both points.
x = 1 → (3) → Minimum, x = 4 → (-3) → Maximum (A1)
y = -x³/3 + 5x²/2 - 4x (+c) (B2, 1, 0) +c not needed. –1 each error or omission.
Uses x = 6, y = 2 in an integrand to find c → c = 8 (M1A1) Statement of the final equation not required.

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Topic: Differentiation, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, Oct/Nov 2017 Paper 1 Variant 2, question 8.

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