My question is about finding the extrema of a multidimensional function, $f:\mathbb{R}^n\rightarrow \mathbb{R}$. From lecture I know that
$H_f(x_0) < 0 $ implies a isolated maximum
$H_f(x_0) > 0 $ implies a isolated minimum
$H_f(x_0)$ indefinite implies a saddle point
Where $H_f(x_0)$ is the Hesse-Matrix at point $x_0$. So how can I identify non isolated maxima? What about the cases where $H_f$ is positive (or negative) semidefinite? What if $H_f = 0$?
I know of a specialized test for the $\mathbb{R^2}$, which answers the above questions, but I wonder about the $\mathbb{R^n}$. What rules can be applied there?