已知x、y、z均为正数,且xyz(x+y+z)=1,那么(x+y)(y+z)的最小值是_.

问题描述:

已知x、y、z均为正数,且xyz(x+y+z)=1,那么(x+y)(y+z)的最小值是______.

(x+y)(y+z)=xy+y2+yz+zx
=y(x+y+z)+zx≥2

y(x+y+z)zx
=2.(当且仅当y(x+y+z)=zx时取等号).
故答案为:2.