Finding Critical Point for function of two variables

The function f(x,y)=8x^3+144xy+8y^3

has exactly two critical points, call them (x1,y1) and (x2,y2).
You are required to find the critical points and classify them using the second-derivative test.

Can someone please show me how to do this? I'm not sure whether you're meant to equate things or put x's and y's on opposite sides of the equation etc. HELP :(

Taking partial derivatives,
f_x = 24x^2 + 144y and f_y = 144x + 24y^2.

Set these equal to 0 to solve for critical points:
24x^2 + 144y = 0 and 144x + 24y^2 = 0
==> y = (-1/6)x^2 and 6x + y^2 = 0

So, 6x + ((-1/6) x^2)^2 = 0
==> 6x + (1/36)x^4 = 0
==> x(216 + x^3) = 0
==> x = 0 or -6.

So, the critical points are (x, y) = (0, 0) and (-6, -6).
-----------------
Now, we classify these with the second derivative test.
f_xx = 48x, f_yy = 48y, f_xy = 144.
==> D = (f_xx)(f_yy) - (f_xy)^2 = (48x)(48y) - 144^2.

Since D(0, 0) < 0, we have a saddle point at (0, 0).
Since D(-6, -6) > 0 and f_xx (-6, -6) < 0, we have a local maximum at (-6, -6).

I hope this helps!