1.
(x-2)/(x+2)>0,x∈(-∞,-2)∪(2,+∞)
f(x)定义域[α,β]包含于(-∞,-2)∪(2,+∞)
α>2或α<-2
f(x)在[α,β]是减函数,0<a<1
值域为[log(a)a(β-1),log(a)a(α-1)],a(α-1)>0,
α>1
∴α>2
2.
f(x)在[α,β]是减函数,0<a<1
值域为[log(a)a(β-1),log(a)a(α-1)]
f(α)=log(a)[(α-2)/(α+2)]=log(a)[a(α-1)]
(α-2)/(α+2)=a(α-1)
f(β)=log(a)[(β-2)/(β+2)]=log(a)[a(β-1)]
(β-2)/(β+2)=a(β-1)
α,β是方程(y-2)/(y+2)=a(y-1)两大于2的不等实根
g(y)=ay^2+(a-1)y-2(a-1)
则需满足下面3个条件:
1)△>0
△=(a-1)^2+8a(a-1)=9a^2-10a+1>0,
(9a-1)(a-1)>0,a<1/9或a>1
2)对称轴在x=2右侧
-(a-1)/(2a)>2,a<1/5
3)g(2)>0
4a+2(a-1)-2(a-1)>0,a>0
∴0<a<1/9
实数a取值范围(0,1/9)
