Basic Analysis I — Real Analysis · Problem 4.3.6

The Derivative — Jiri Lebl

Mathematics — Real Analysis · taylor-remainder-limit-iff

Problem

Suppose

f : [a,b] \to \mathbb{R}

has

n+1

continuous derivatives and

x_0 \in (a,b)

. Prove:

f^{(k)}(x_0) = 0

for all

k = 0, 1, 2, \ldots, n

if and only if

\displaystyle\lim_{x \to x_0} \frac{f(x)}{(x - x_0)^{n+1}}

exists.