Question

(本小題滿分10分)
已知函數(shù)，當(dāng)點(diǎn)(x，y) 是函數(shù)y = f (x) 圖象上的點(diǎn)時(shí)，點(diǎn)是函數(shù)y = g(x) 圖象上的點(diǎn)．
(1)   寫出函數(shù)y = g (x) 的表達(dá)式；
(2)   當(dāng)g(x)-f (x)0時(shí)，求x的取值范圍；
(3)   當(dāng)x在(2) 所給范圍內(nèi)取值時(shí)，求的最大值．

Answer 1

（1）
（2）
（3）[g(x)-f (x)]_max=

解得0 ≤ x ≤ 1；·········································································· 6分
(3) 當(dāng)0≤ x≤ 1時(shí)，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(dāng)k=時(shí)，x=∈[0，1]，
∴當(dāng)x=時(shí)，[g(x)-f (x)]_max=．········································· 10分