设x,y∈R,比较x^2+y^2+1与x+y+xy

题目

设x,y∈R,比较x^2+y^2+1与x+y+xy

答案

因为(x^2+y^2+1)-(x+y+xy)=x^2+y^2+1-x-y-xy=1/2*(2x^2+2y^2+2-2x-2y-2xy)=1/2*[(x^2-2xy+y^2)+(x^2-2x+1)+(y^2-2y+1)]=1/2*[(x-y)^2+(x-1)^2+(y-1)^2]又因为(x-y)^2≥0且(x-1)^2≥0且(y-1)^2≥0,所以(x-y)^2+(x-1)^...