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Given f:R^2->R, f(x,y)=x^4+y^4-2(x-y)^2

1-Prove that (sqrt(2),-sqrt(2)) and (-sqrt(2),sqrt(2)) are absolute minimums

2-are there any local maximums?

1-I found out that the critical points lie on the line y=-x, and i suppose i should prove that f(sqrt(2),-sqrt(2))=-8<f(x,y) for every (x,y) but i dont know how to do this.

2-I found that there arent any local maximums, but i would like you to correct me if i am wrong.

Thank you very much, Paul.