Is $(0,0)$ a saddle point for the given function?

Question

I need to find the critical points for the function

$f(x) = 3(x^2 + y^2) - 2(x^3 - y^3) + 6xy$ and also test whether they are maxima/minima/saddle point.

Now the only critical point is (0,0)

however at (0,0) $rt - s^2 =0$ then second derivative test fails,

If I take the line $y = -x$ then

$f(x, -x) = -4x^3$ then clearly for along the neighborhood of $(0,0)$ $f$ has both positive and negative values .

Hence , $(0,0)$ is a saddle point .

Is my solution and answer correct ? Can someone please verify ?

Thank you.

Answer 1

The question has already been answered in the comments, I am just answering for the sake of completeness so that it does not remain unsolved.

The process is correct and $(0,0)$ is the saddle point for the given function.