Finding Extrema of $f(x,y)=x^4+y^4-4xy$ Let $f(x,y)=x^4+y^4-4xy$ 
How do I find all the relative extrema and saddle points of $f$ which lie within the open square ${(x,y) | -2<x<2,-2<y<2}$. And also if $f$ was in the closed region ${(x,y) | -2\le x\le 2,-2\le y\le 2}$ what would be the absolute maxima and minima of $f$ on this closed region and where do these extreme values occur?
What I have Tried:
I first set $f_1=4(x^3-y)$ and $f_2=4(y^3-x)$ I don't know what to do next though.
 A: Hint: Look at the corresponding closed region $R$, so that the extrema will either be critical points in int$R$ or they will lie on the boundary of $R$. For the saddle points, use the second derivative test.
Second hint: 
Step 1:
Solve the following equations and use the points in the second derivative test to decide if they are local extrema or saddle points. Suggestion: let $z=y/x$
$f_{x}=4x^{3}-4y=0\\f_{y}=4y^{3}-4x=0$
Step 2:
Now consider each piece of the boundary, making the appropriate substitution and extremize the function (of a single variable) you get in each case. For example, if we look at the top side of the square, where $y=2$ and $-2<x<2$, we get 
$f(x,2)=x^{4}-8x+16$, from which $f'(x,2)=4x^{3}-8=0$ gives $x=2^{1/3}$ as a critical point. Now determine what type of critical point it is, substitute into $f(x,y)$ and compare it with what you got in step 1. Then check $x=2, x=-2$ in $f(x,2)$ and write down the result. 
repeat for the other sides of the square.  
All told, you get a table of values from which you can read off the maxima and minima. 
A: use the two equations as simultaneous equations at zero and solve the homogeneous equations. $4(x^3 - y) = 0$ and $4(y^3 - x) = 0$. Pretty sure that this gives a line in space rather than point that you might be use to. this is because there is no positive numbers without variables 
ie: $4(x^3 - y) = 3$ and $4(y^3 - x) = 5$
will give a single point extrema.
A: you can use Lagrangians ( Constrained Optimization). I give you an example here : 
http://mat.gsia.cmu.edu/classes/QUANT/NOTES/chap4.pdf
