Date | May 2021 | Marks available | 2 | Reference code | 21M.2.SL.TZ2.5 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 2 |
Command term | Find | Question number | 5 | Adapted from | N/A |
Question
The functions f and g are defined for x∈ℝ by f(x)=6x2-12x+1 and g(x)=-x+c, where c∈ℝ.
Find the range of f.
Given that (g∘f)(x)≤0 for all x∈ℝ, determine the set of possible values for c.
Markscheme
attempting to find the vertex (M1)
x=1 OR y=-5 OR f(x)=6(x-1)2-5
range is y≥-5 A1
[2 marks]
METHOD 1
(g∘f)(x)=-(6x2-12x+1)+c (=-(6(x-1)2-5)+c) (A1)
EITHER
relating to the range of f OR attempting to find g(-5) (M1)
5+c≤0 (A1)
OR
attempting to find the discriminant of (g∘f)(x) (M1)
144+24(c-1)≤0 (120+24c≤0) (A1)
THEN
c≤-5 A1
METHOD 2
vertical reflection followed by vertical shift (M1)
new vertex is (1, 5+c) (A1)
5+c≤0 (A1)
c≤-5 A1
[4 marks]