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## Main Question or Discussion Point

If ##f'(0) = 0## and ##n## is the smallest natural number such that ##f^{(n)}(0)\neq 0##, then the higher-order derivative test states the following:

1. If ##n## is even and ##f^{(n)}(0)>0##, then ##f## has a local minimum at ##0##.

2. If ##n## is even and ##f^{(n)}(0)<0##, then ##f## has a local maximum at ##0##.

3. If ##n## is odd, then ##f## has an inflection point at ##0##.

4. If ##f^{(n)}(0)=0## for all ##n##, then the higher-order derivative test is inconclusive.

I'd like to find a method of finding minima and maxima in the case where the higher-order derivative test is inconclusive, using the concept of germs. Let ##X## be the set of all functions infinitely differentiable at ##0## for which ##f^{(n)}(0)=0## for all ##n##. Let's define an equivalence relation on $X$ by saying that ##f\sim g## if there exists a sufficiently small open interval ##I## containing ##0## such that ##f(x)=g(x)## for all ##x## in ##I##. Then the set of germs of ##X## denotes the set ##Y## of equivalence classes of elements of ##X## under this equivalence relation. (Note that ##Y## has infinitely many elements, as two functions which have all their nth derivatives in common can have completely different germs.)

My question is, does there exist a nontrivial function ##F## from ##Y## to ##\mathbb{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##?

1. If ##n## is even and ##f^{(n)}(0)>0##, then ##f## has a local minimum at ##0##.

2. If ##n## is even and ##f^{(n)}(0)<0##, then ##f## has a local maximum at ##0##.

3. If ##n## is odd, then ##f## has an inflection point at ##0##.

4. If ##f^{(n)}(0)=0## for all ##n##, then the higher-order derivative test is inconclusive.

I'd like to find a method of finding minima and maxima in the case where the higher-order derivative test is inconclusive, using the concept of germs. Let ##X## be the set of all functions infinitely differentiable at ##0## for which ##f^{(n)}(0)=0## for all ##n##. Let's define an equivalence relation on $X$ by saying that ##f\sim g## if there exists a sufficiently small open interval ##I## containing ##0## such that ##f(x)=g(x)## for all ##x## in ##I##. Then the set of germs of ##X## denotes the set ##Y## of equivalence classes of elements of ##X## under this equivalence relation. (Note that ##Y## has infinitely many elements, as two functions which have all their nth derivatives in common can have completely different germs.)

My question is, does there exist a nontrivial function ##F## from ##Y## to ##\mathbb{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##?