设x>=1,y>=1,证x+y+1/xy

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设x>=1,y>=1,证x+y+1/xy
设x>=1,y>=1,证x+y+1/xy<=1/x+1/y+xy.

设x>=1,y>=1,证x+y+1/xy
证明：因为x≥1,y≥1,所以(xy－1)(x－1)(y－1)≥0,
展开得x^2y^2－x^2y－xy^2＋x＋y－1≥0,移项得：x^2y＋xy^2＋1≤x^2y^2＋x＋y.
两边同除以xy得x＋y＋1/xy≤1/x＋1/y＋xy

x+y+1/xy=(x^2y+xy^2+1)/(xy)
1/x+1/y+xy=(y+x+x^2y^2)/(xy)
y+x+x^2y^2-(x^2y+xy^2+1)=（xy-1)(x+y-1)>=0
因此x+y+1/xy=(x^2y+xy^2+1)/(xy)＜＝(y+x+x^2y^2)/(xy)＝1/x+1/y+xy

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