How do I classify this critical point? I am given the function $ f(x,y) = x^4 + y^3 $ and I am asked what happens at input $(0,0)$. After finding the first derivative and the Hessian Matrix; they are both 0 matrices. I know then that it is a critical point but since the determinant of the Hessian is $0$, I don't really know what to do. The question asks to use other reasoning to find the answer. 
I know that it looks like a minimum with respect to $x$ and a point of inflection with respect to $y$. (Plotted them separately) and I am just not sure how to classify it. 
I am asked not to use Lagrange multipliers. This is part of my Homework for a freshman Multivariable Calc. and Linear Algebra class. See this 3d Plot of the function
 A: It is a similiar situation when you want to look at a single-variable function and $f'(t) = f''(t) = 0$: you look at higher derivatives. If $f\colon \Bbb R^n \to \Bbb R$, we have the Taylor formula $$f(p+\vec{h})-f(p) = \sum_{k=1}^n \frac{1}{k!}D^kf(p)(\vec{h})^k + o(\vec{h}),$$where $o(\vec{h})/\|\vec{h}\|^{n} \to 0$. In our case, the third derivative (a trilinear map) is given by (if $\vec{h} = (h_1,h_2)$)$$D^3f(0,0)(\vec{h})^3 = 6 (h_2)^3,$$so we conclude that $(0,0)$ is an inflection point, since $$f(h_1,h_2) = (h_2)^3 + o(\vec{h}),$$and the sign of $f(h_1,h_2)$ is the same of $(h_2)^3$ for small enough $\vec{h}$, and we can pick small errors $\vec{h}$ with $h_2$ having both signs.

If $f\colon \Bbb R^n \to \Bbb R$, then the third derivative $D^3f(p)$ is defined by $$D^3f(p)(v,w,z) = \sum_{i,j,k=1}^n \frac{\partial^3 f}{\partial x_i \partial x_j \partial x_k}(p) v_iw_jz_k,$$where $v = (v_1,\ldots,v_n)$ and similarly for the other two vectors.
A: It is either a maximum, minimum or saddle point right? It cannot be a maximum or a minimum, take a look at:
\begin{align}
f(0,y) = y^3
\end{align}
You can find points that are less than $0$ and points that are greater than $0$ in each neighbourhood of $(0,0)$, so it cannot be a maximum or a minimum. Therefore it is a saddle point.
