已知函数f(x)=a(x-1)/x-2,a为常数.

1
a(x-1)/(x-2) -2 >0
两边同乘(x-2)^2
a(x-1)(x-2) -2(x-2)^2 >0
(ax-a -2) (x-2) >0
方程(ax-a -2) (x-2)=0的两根为 1+2/a,2
所以 1+2/a = 3; 2/a = 2; a=1
2
x∈（2,+∞）,x-1>0,x-2>0
a(x-1)/(x-2) < (x-3)
a < (x-3)(x-2)/(x-1)
= (x-3)(x-1-1)/(x-1)
= (x-3)(x-1)/(x-1) - (x-3)/(x-1)
= x-3 - 1 +2/(x-1)
= (x-1) + 2/(x-1) -3
当(x-1) = 2/(x-1),即 x= 1+√2,右边(x-1) + 2/(x-1) -3有最小值 2√2 -3
所以
a