Question

(本小題滿分10分)
已知函數，當點(x，y) 是函數y = f (x) 圖象上的點時，點是函數y = g(x) 圖象上的點．
(1)   寫出函數y = g (x) 的表達式；
(2)   當g(x)-f (x)0時，求x的取值范圍；
(3)   當x在(2) 所給范圍內取值時，求的最大值．

Answer 1

（1）
（2）
（3）[g(x)-f (x)]_max=

解得0 ≤ x ≤ 1；·········································································· 6分
(3) 當0≤ x≤ 1時，g(x) -f (x) =-log ₂(x+1)，
令，
則kx² + (2k-3)x + (k-1) = 0，························································· 7分
∵x的取值存在，∴D = (2k-3)²-4k(k-1) ≥ 0，
解得：，·········································································· 9分
當k=時，x=∈[0，1]，
∴當x=時，[g(x)-f (x)]_max=．········································· 10分