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 .