I have this function and I am looking for the stationary points:

$\displaystyle

f(x,y) = x^4 + 2x^2y^2 + y^4 + 4x^2 - 4y^2 + 6

$

I started by trying to solve the two partial derivatives:

$\displaystyle

4x(x^2 + y^2 + 1) = 0, (E.1)

$

$\displaystyle

4y(y^2 + x^2 - 1) = 0, (E.2)

$

If $\displaystyle x = 0$, then $\displaystyle (E.2)$ becomes:

$\displaystyle

4y(y^2 - 1) = 0

$

Thus $\displaystyle y = 0$ , $\displaystyle y = 1$ , or $\displaystyle y = -1$, so there are three stationary points at $\displaystyle (0,0)$ , $\displaystyle (0,1)$ , and $\displaystyle (0,-1)$ , correct?

Next, if:

$\displaystyle

x^2 + y^2 + 1 = 0,

$

i.e.

$\displaystyle

y^2 = -x^2 - 1

$

then $\displaystyle (E.2)$ becomes... uh...

$\displaystyle

-8y = 0

$

this is where I run into trouble.

Is $\displaystyle x \ge 0$ ?

How do I go about finding other points? Are there other stationary points?