Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3 $ and classify their type.
For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted here, because 0 is neither + nor -, can't we conclude immediately that $(0,0)$ must be a critical point? Why does the solution calculate $D_2$?
I forgot to define $D_j$. It's just the upper left j x j submatrix, the principal minor.
In dimension $2$, if the Hessian matrix is $\begin{pmatrix}0&b\\b&d\end{pmatrix}$ with $b\ne0$ then the eigenvalues are $\frac12(d\pm\sqrt{d^2+4b^2})$, one positive and one negative, hence the critical point is a saddle point. If the Hessian matrix is $\begin{pmatrix}0&0\\0&d\end{pmatrix}$, the eigenvalues are $0$ and $d$ hence the critical point is not a saddle point.
To sum up, the fact that the $(1,1)$ entry of the Hessian matrix at a critical point is $0$ is almost, but not quite, sufficient to declare that this point is a saddle point.