# Higher derivative test is conclusive for locally analytic function

From Calculus

This article describes a situation, or broad range of situations, where a particular test or criterion isconclusive, 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 is a function, is a point in the interior of the domain of , and is analytic about , i.e., there is a power series centered at that converges to on an open interval containing . In particular, this means that is infinitely differentiable at . The assumption of being a critical point also forces .

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

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