若xy=2x+y,求x+y的取值范围

题目

若xy=2x+y,求x+y的取值范围

答案

y=2x/(x-1)
x+y=x+2x/(x-1)
=(x^2-x+2x)/(x-1)
=(x^2+x)/(x-1)
=[(x-1)^2+3(x-1)+2]/(x-1)
=(x-1)+2/(x-1)+3
所以x+y>=3+2根号2,当且仅当x=1+根号2时等号成立
或x+y