Higher derivative test is conclusive for locally analytic function

This article describes a situation, or broad range of situations, where a particular test or criterion is conclusive, i.e., it works as intended to help us determine what we would like to determine.
The test is higher derivative test. See more conclusive cases for higher derivative test | inconclusive cases for higher derivative test

Statement

Suppose $f$ is a function, $c$ is a point in the interior of the domain of $f$, and $f$ is analytic about $c$, i.e., there is a power series centered at $c$ that converges to $f$ on an open interval containing $c$. In particular, this means that $f$ is infinitely differentiable at $c$. The assumption of $c$ being a critical point also forces $f'(c)=0$.

In this case, the higher derivative test must always be conclusive for $f$ at $c$.

In particular, the result applies to polynomial functions and rational functions.