extremes

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 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.

matt grime

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

pbialos

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.