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^2

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