Basic Analysis I — Real Analysis · Problem 4.3.4

The Derivative — Jiri Lebl

Mathematics — Real Analysis · taylor-polynomial-bounding

Problem

Suppose

f : \mathbb{R} \to \mathbb{R}

has

n

continuous derivatives. Show that for every

x_0 \in \mathbb{R}

, there exist polynomials

P

and

Q

of degree

n

and an

\epsilon > 0

such that

P(x) \leq f(x) \leq Q(x)

for all

x \in [x_0, x_0 + \epsilon]

and

Q(x) - P(x) = \lambda(x - x_0)^n

for some

\lambda \geq 0