已知正实数x，y，z满足2x（x+1y+1z）=yz，则（x+1y）（x+1z）的最小值为22．

Question

已知正实数x，y，z满足2x（x+(1)/(y)+(1)/(z)）=yz，则（x+(1)/(y)）（x+(1)/(z)）的最小值为

√(2)

√(2)

．

Answer 1

∵正实数x，y，z满足2x（x+(1)/(y)+(1)/(z)）=yz，
∴x²+x（(1)/(y)+(1)/(z)）=(1)/(2)yz，
∴（x+(1)/(y)）（x+(1)/(z)）=x²+x（（(1)/(y)+(1)/(z)）+(1)/(yz)
=(1)/(2)yz+(1)/(yz)≥2√((1)/(2))=√(2)．
当且仅当yz=√(2)，取得最小值√(2)．
故答案为：√(2)．