Question

若xy=2x+y,求x+y的取值范围

Answer 1

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<=3-2根号2，当且仅当x=1-根号2时等号成立

Answer 2

由xy=2x+y得y=-2x/(1-x)(x不等于1)
当x=1时，xy=2x+y式不成立，从而x不等于1
x+y==x+-2x/(1-x)
当x趋向正无穷时，上式趋向负无穷
当x趋向负无穷时，上式趋向负正穷

Answer 3

用柯西不等式。xy=2x+y相当于(2/x)+(1/y)=1，x+y=(x+y)*((2/x)+(1/y))>=(√2+1)²