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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?

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    @ftiaronsem: there is a criterion involving higher derivatives to handle some cases wherw $H_f$ is zero. It can be found in oldish calculus books (I can only think of Rey Pastor's *Análisis Matemático* right now, which is in Spanish...) There is no 100% effective criterion to decide which only involves derivatives (of any order) at a point: one can construct examples of functions all of whose derivatives vanish at a point and which have there a maximum, or a minimum, or whatever you like.2011-06-16

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Assume that $f$ is of class $C^2$ in a ball $B_r(x_0)$, and 1) $\nabla f(x_0) = 0$, 2) $H_f(x) \geq 0$ for every $x\in B_r(x_0)$. Then $x_0$ is a local minimum. Indeed the second assumption implies that $f$ is convex in $B_r(x_0)$, so that condition 1) is sufficient in order to have a minimum point at $x_0$. (A similar statement holds true if the Hessian matrix is negative semidefinite in a neighborhood of $x_0$.)

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    You are right, a whole neighborhood will do. It is just possibly not a strict local minimum. If a function is convex, then the tangent plane at a point is below (or on) the graph.2011-06-16