Let X be the set of all functions f from R to R infinitely differentiable at 0 and for which f^(n)(0)=0 for all n (i.e. the case where the higher-order derivative test is inconclusive.) Let Y be the set of germs of X, i.e. the set of equivalence classes of elements of X under the equivalence relation ~ defined as follows: f ~ g if there exists an open interval I containing 0 such that f(x) = g(x) for all X in I.
My question is, does there exist a nontrivial function F from Y to R such that if F evaluated at a particular germ yields a positive number, then all the functions in the germ have a local minimum at 0, and if it yields a negative number then all the functions in the germ have a local maximum at 0? By nontrivial I mean that I want F to generally assign different values to different germs, although it’s OK if rarely two germs coincidentally have the same value of F. (Just like if you pick two random functions they almost certainly won’t have the same derivative at 0, but on rare occasion they might.)
My question is, does there exist a nontrivial function F from Y to R such that if F evaluated at a particular germ yields a positive number, then all the functions in the germ have a local minimum at 0, and if it yields a negative number then all the functions in the germ have a local maximum at 0? By nontrivial I mean that I want F to generally assign different values to different germs, although it’s OK if rarely two germs coincidentally have the same value of F. (Just like if you pick two random functions they almost certainly won’t have the same derivative at 0, but on rare occasion they might.)
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