题目
xy都大于等于1,求证x+y+1/xy小于等于1/x+1/y+xy
提问时间：2021-03-23
答案
x+y+1/(xy)-1/x-1/y-xy
=(x²y+xy²+1-y-x-x²y²)/(xy)
=[(x²y-x)+(xy²-y)-(x²y²-1)]/(xy)
=[x(xy-1)+y(xy-1)-(xy+1)(xy-1)]/(xy)
=(xy-1)(x+y-xy-1)/(xy)
=(xy-1)[(x-xy)+(y-1)]/(xy)
=(xy-1)[-x(y-1)+(y-1)]/(xy)
=(xy-1)(y-1)(1-x)/(xy)
由x>1 y>1 得xy>1 xy-1>0；y-1>0；1-x0
(xy-1)(y-1)(1-x)/(xy)
=(x²y+xy²+1-y-x-x²y²)/(xy)
=[(x²y-x)+(xy²-y)-(x²y²-1)]/(xy)
=[x(xy-1)+y(xy-1)-(xy+1)(xy-1)]/(xy)
=(xy-1)(x+y-xy-1)/(xy)
=(xy-1)[(x-xy)+(y-1)]/(xy)
=(xy-1)[-x(y-1)+(y-1)]/(xy)
=(xy-1)(y-1)(1-x)/(xy)
由x>1 y>1 得xy>1 xy-1>0；y-1>0；1-x0
(xy-1)(y-1)(1-x)/(xy)
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