# Is the point at $(0,0)$ of the graph $z=x^3 + y^3$ considered a saddle point?

Is the point at $(0,0)$ of the graph $z=x^3 + y^3$ considered a saddle point?

I was given that function, and I used the second-order derivative test only to find that the $Hf(0,0) = 0$, with the critical point being at 0,0. Is anything that is not a local minimum or a local maximum considered a saddle point? What the graph looks like

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3. If I am standing at (0,0,0) and you give me an $\epsilon$, I can step $(\epsilon,0)$ and make the function positive or $(-\epsilon,0)$ and make the function negative. So, no matter how small epsilon you give me, I can always move to points that increase the function value and reduce it. This makes the (0,0,0) a saddle point.
Standard example of a local min with vanishing Hessian is $x^4 + y^4$. The common definition of saddle point is that it is neither a local min nor local max, and this is defined without any reference to derivatives or Hessian. – Erick Wong Jan 16 '13 at 5:31