A function f is defined, for all real x, by f(x) = x - x². Find the greatest value of f(x) and the value of x for which this occurs.

Mark scheme

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

1. [M1] Completing the square as far as e.g. constant - (x - 1/2)² or calculus as far as 1 - 2x = 0 or finding roots x = 0 and x = 1 and using symmetry soi
2. [A1] when x = 1/2 Implies M1 if not clearly from wrong working
3. [B1] [greatest value =] 1/4
4. [B1] Valid comment e.g. when x ≥ 1, f' is always decreasing Allow e.g. a sketch with a comment such as the curve is one-one [when x ≥ 1] or e.g. the curve is one-one when x > 1/2
5. [M1] k(10) = 8 or 5+√(10-1)=8 or stating h(8) or [hk(x) =] lg(7+√x−1)
6. [A1] h(8) = 1 or lg(8+2)=1 cao [hk(10) =] lg(7 + √10−1)=1
7. [M1] (y-5)² = x-1 or (x-5)² = y-1
8. [A1] k⁻¹(x)=(x-5)²+1 isw or k⁻¹(x) = x² -10x+26 isw
9. [B1, B1] 5<x<15 B1 for 5 < x oe and B1 for x < 15 oe allow (5, 15); one mark for each limit of the interval; if B0 then SC1 for 5 ≤ x ≤ 15 or '5 to 15' or [5, 15] etc.
10. [B1] 1<k⁻¹(x)<101 allow (1, 101)