Real function/Extrema/Higher derivatives/Fact/Proof

Proof

Under the given conditions, the Taylor formula becomes

with some (depending on ) between and . Depending on whether or holds, we have (due to the continuity of the -th derivative) or for , for a suitable . For these , we have , so that the sign of depends on the sign of .
For even, is odd and therefore the sign of changes at (for , the sign is negative, and for , the sign is positive). Since the sign of does not change, the sign of is changing. This means that there can not be an extremum.
Suppose now that is odd. Then is even, hence for all in the neighborhood. This means, in the neighborhood, in case , that holds, and we have an isolated minimum in . If , then holds, and we have an isolated maximum in .