First 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 is conclusive, i.e., it works as intended to help us determine what we would like to determine.
The test is first derivative test. See more conclusive cases for first derivative test | inconclusive cases for first derivative test
Contents
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 first derivative test is conclusive for .
Related facts
- Higher derivative test is conclusive for locally analytic function
- First derivative test is conclusive for function with algebraic derivative (this is a somewhat weaker version because having an algebraic derivative implies being locally analytic)
Facts used
- Function analytic about a point has isolated zeros near the point
- First derivative test is conclusive for differentiable function at isolated critical point
Proof
The proof follows directly by applying Fact (1) to the derivative and combining with Fact (2).