The function f(x) is given by f(x) = (3x³-1)/(x³+1) for 0 ≤ x ≤ 3.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

1. [M1] (i) f'(x)= ((x³ +1)9x² − (3x³ −1)3x²)/(x³ +1)² quotient rule or product rule
2. [A1] all correct
3. [A1] = 12x²/(x³ +1)² www beware 9x^6-9x^6 gets A0
4. [M1] (ii) ∫ x²/(x³+1)² dx = (1/(-12)) * (3x³-1)/(x³+1) c x (3x³-1)/(x³+1)
5. [A1] FT c = -1/12 (their 12)
6. [DM1] [(1/12) * (3x³-1)/(x³+1)]_1^2 top limit - bottom limit in their integral.
7. [A1] = 7/54 or 0.130 or 0.1296 or 0.12
8. [B1] (iii) x = (3y³-1)/(y³+1) make y³ or x³ the subject
9. [B1] f⁻¹(x) = ³√((x+1)/(3-x)) FT take cube root (as long as y³ or x³ equals a fraction with terms in x or y only) oe
10. [B1] FT change x and y – can be done at any time
11. [B1] Domain : -1 ≤ x < 20/7 Allow upper limit of 2.86. Do not isw