Given a smooth function of several variables, whose first derivatives vanish at the origin. Suppose the matrix of second derivatives is degenerate at the origin. For example all the second derivatives vanish.

What are the ways of classical Calculus to check whether this is a min/max/saddle? (Some non-calculus ways?)

For example, is the origin min/max/saddle for $f(x,y)=x^{10}+x^2y^2+y^{10}-10000xy^8$?

Sometimes a case can be checked by a locally analytic change of variables, i.e. in a constructive manner. In most cases the needed change of variables is a local homeomorphism, i.e. smth non-constructive.

Singularity theory provides some invariants that sometimes help to answer this question. (The simplest such invariant: the Newton diagram.) I do not know any general method to attack these problems.


upd. (added on 23.12.2013) In this paper Vasil'iev shows: among the (analytic) functions with the given Newton diagram (at the origin) there is a function with the (strict) minimum at the origin iff all the vertices of the Newton diagram have only even coordinates.



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