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|f'(x)| ≤ C if |f(x)| ≤ A, |f''(x)| ≤ B

Source: 1963 Swedish Mathematical Competition p6

March 21, 2021

analysisalgebrainequalitiesfunctionFunctional inequality

Problem Statement

The real-valued function

f(x)

is defined on the reals. It satisfies

|f(x)| \le A

|f''(x)| \le B

for some positive

A, B

(and all

x

). Show that

|f'(x)| \le C

, for some fixed

C

, which depends only on

A

and

B

. What is the smallest possible value of

C