Basic Analysis I — Real Analysis · Problem 4.3.7

The Derivative — Jiri Lebl

Mathematics — Real Analysis · ode-second-order-uniqueness

Problem

Suppose

a, b, c \in \mathbb{R}

and

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

is differentiable,

f''(x) = a

for all

x

f'(0) = b

, and

f(0) = c

. Find

f

and prove that it is the unique differentiable function with this property.