已知x,y都大于等于1,求证：x+y+1/xy=
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已知x,y都大于等于1,求证：x+y+1/xy=
已知x,y都大于等于1,求证：x+y+1/xy=
已知x,y都大于等于1,求证：x+y+1/xy=
x>=1,y>=1
则(x-1)(y-1)>=0,1>=1/(xy)>0
(x-1)(y-1)>=(x-1)(y-1)/(xy)
xy-x-y+1>=1-1/x-1/y+1/(xy)
x+y+1/(xy)<=1/x+1/y+xy
x>=1,y>=1
则(x-1)(y-1)>=0,1>=1/(xy)>0
(x-1)(y-1)>=(x-1)(y-1)/(xy)
xy-x-y+1>=1-1/x-1/y+1/(xy)
x+y+1/(xy)<=1/x+1/y+xy
两式同乘xy (x+y+1/xy)*xy=x^2*y+x*y^2+1
(1/x+1/y+xy)*xy=y+x+(xy)^2
当x=y时
x^2*y+x*y^2=2*x^3
(xy)^2=x^4
当x=y=1时
2*x^3 =2，x^4=1
...
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两式同乘xy (x+y+1/xy)*xy=x^2*y+x*y^2+1
(1/x+1/y+xy)*xy=y+x+(xy)^2
当x=y时
x^2*y+x*y^2=2*x^3
(xy)^2=x^4
当x=y=1时
2*x^3 =2，x^4=1
x^2*y+x*y^2+1=2*x^3 +1=2+1=3
y+x+(xy)^2=1+1+x^4=3
两式相等
当x=y>1时
2*x^3
x^2*y+x*y^2=xy*(x+y)
xy>x+y
x^2*y+x*y^2<(xy)^2
[(x+y+1/xy)*xy=x^2*y+x*y^2+1]<[(1/x+1/y+xy)*xy=y+x+(xy)^2]
即：x+y+1/xy=<1/x+1/y+xy
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