A-maths

(a) f(x) = -x^2 + kx
= -(x^2 - kx + k^2/4) + k^2/4
= -(x - k/2)^2 + k^2/4
The minimum value of f(x) occurs when the square term is a minimum, i.e. 0
The minimum is k^2/4 when x = k/2
(b) g(x) = y = 2x ... (1)
f(x) = y = -x^2 + kx ...(2)
Sub (1) into (2), 2x = -x^2 + kx
x^2 + (2 - k)x = 0
x[x + (2-k)] = 0
x = 0 or x = k-2
when x = 0, y = 0
when x = k-2, y = 2k -4
the intersection points are (0,0) and (k-2, 2k-4)
(c) (i)http://img37.imageshack.us/img37/1720/paray.jpg


圖片參考：http://img37.imageshack.us/img37/1720/paray.jpg
(ii) The range when f(x) >= g(x) is [0,4]
The greatest value of f(x) in this range is 6^2/4 = 9
(d) When k = 3, the range when f(x) >= g(x) is [0,3-2] = [0,1]
The greatest value of f(x) in this range is 3^2/4 = 2.25