(x+1/x)(y+1/y)=xy+1/xy+x/y+y/x=xy+1/xy+(x^2+y^2)/xy
=xy+1/xy+[(x+y)^2-2xy]/xy=xy+1/xy+(1-2xy)/xy
=xy+2/xy-2
设t=xy
xy≤(x+y)^2/4=1/4
所以0≤t≤1/4
t+2/t在(0,√2]上递减,在[√2,+∞)上递增
(x+1/x)(y+1/y)=xy+2/xy-2≥1/4+8-2=25/4
证明：若x>0,y>0,x+y=1,则（x+1/x)(y+1/y)≥25/4
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