设x>0,y>0,且(x-1)(y-1)≥2,则xy的取值范围为?

题目

设x>0,y>0,且(x-1)(y-1)≥2,则xy的取值范围为?

答案

(x-1)(y-1)≥2
xy-(x+y)+1≥2
xy≥1+x+y 1)
xy≥1+√(x^2+y^2+2xy)
xy-1≥√(x^2+y^2+2xy)
(xy)^2-2xy+1≥x^2+y^2+2xy>=4xy
(xy)^2-6xy+1≥0
xy≥3+2√2或xy≤3-2√2
由1）得xy>1
所以xy≥3+2√2