（不等式选讲选做题）x、y＞0，x+y=1，则(x+1x)(y+1y)的最小值为．

Question

（不等式选讲选做题）x、y＞0，x+y=1，则(x+(1)/(x))(y+(1)/(y))的最小值为______．

Answer 1

(x+(1)/(x))(y+(1)/(y))＝xy+(x)/(y)+(y)/(x)+(1)/(xy)=(x2y2+x2+y2+1)/(xy)
=(x2y2+(x+y)2−2xy+1)/(xy)
=xy+(2)/(xy)−2
∵x+y=1
∴xy≤((x+y)/(2))2＝(1)/(4)
令xy=t则0＜t≤(1)/(4)
∴(x+(1)/(x))(y+(1)/(y))＝t+(2)/(t)−2
∵t+(2)/(t)−2在(0，(1)/(，4)]递减
当t＝(1)/(4)最小为(25)/(4)
故答案为(25)/(4)