Higher derivative test is conclusive for locally analytic function

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.