Prove that f(sqrt(2),-sqrt(2))=-8<f(x,y)

[pbialos]

I have an exercise i would really appreciate if you could help me with:

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 don't 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.

 

[matt grime]

2. what are the partial derivatves, where do they vanish and what is the discrominant there? post the working.

 

[pbialos]

my work

The partial derivatives are: Fx=4x^3-4(x-y) and Fy=4y^3+4(x-y)
they vanish at x=-y and the discriminant there is 144y^4-96y^2 if my calculations are correct.
Fx and Fy are the partial derivatives with respect to x and y respectively.
What i did next, was to say that if there is a relative maximum at (x,-x):

(a)the discriminant has to be positive and Fxx>0 or
(b)the discriminant has to be equal to 0.

if (a) happens it would mean that Fxx=12x^2-4>0 and discriminant=144y^4-96y^2>0 which is not possible for any point over y=-x.

(b) can only happen if (x,y)=(0,0) but i can prove that it is a saddle point approaching from different directions to (0,0).

So i concluded that there are not relative maximums, is this correct?

If you ask me, i think i can explain it a little bit clearer.
Thank you for your help, Paul.