已知X>=1,y>=1,证明x+y+1/xy

题目

已知X>=1,y>=1,证明x+y+1/xy

答案

1/x+1/y+xy -（ x+y+1/xy）
=1/X - x +1/Y -y +XY -1/xy
=1/xy（y-xxy+x-xyy+xxyy-1）
=1/xy (y+x -xy(y+x) + (xy)^2 -1)
=1/xy ((y+x)(1-xy)+(xy-1)(xy+1))
=1/xy ((xy-1)( xy +1 - y - x))
=(xy-1)/xy *( (x-1)(y-1))
=1/xy * (x-1)(y-1)(xy-1)
x≥1 y≥1 xy≥1
所以上式≥0
所以 右边≥左边 即
x+y+1/xy