Consider the function $f(x,y) = x^{3} -3xy^{2}, x,y \in \Bbb{R}$. Then what can you conclude about its critical point $(0,0) ?$

For this function $f\_{xx} f\_{yy} - f\_{xy}^{2} = 0$ at $(0,0)$ 
So the test gives no result. 

The function can be written as $x(x^{2} - 3y^{2}) = x(x- √3 y)(x+√3 y)$

Now how do we know the nature of function near $(0,0) $ ?