Homework Help: Finding Min/Max Saddle points of 2-var function

1. Oct 15, 2008

1. The problem statement, all variables and given/known data
Find local extrema and saddle points (if any) of the function $f(x,y)=x^4+y^4-4xy-10$

3. The attempt at a solution

Taking the partials and setting them equal to zero I find:

$f_x=4x^3-4y=0\Rightarrow x^3-y=0$ (1)
$f_y=4y^3-4x=0\Rightarrow y^3-x=0$ (2)

Solving (1) for y yields $y=x^3$

Plugging into (2) yields $(x^3)^3-x=0$

$\Rightarrow x(x^8-1)=0$

Now what? I know that x=0 is one solution.

Should I separate the expression in the brackets in to the difference of two squares?

I am reluctant since that will yield $x(x^4-1)(x^4+1)=0$
$\Rightarrow x(x^2-1)(x^2+1)(x^4+1)=0$
$\Rightarrow x(x-1)(x+1)(x^2+1(x^4+1)=0$

and what the crap good is this?

I know x={0,1,-1} and it looks like they just have a bunch of multiplicities.

I guess that does work....so maybe I don't have any other question except:

Is this the right wat to go about solving this?

Casey