Let us find the domain and range of f(x,y)=x2+y22xy. Since x2+y2=0 implies (x,y)=(0,0), we conclude that the domain is R2∖{(0,0)}. Taking into account that (x−y)2≥0, we conclude that x2−2xy+y2≥0, and hence x2+y2≥2xy. It follows that x2+y22xy≤1. On the other hand, (x+y)2≥0 implies x2+y22xy≥−1. It follows that the range of f is [−1,1].
