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Easy gronwall type inequality

Source: 2018 South Korea USCM P6

August 14, 2020

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Problem Statement

Suppose a continuous function

f:[0,1]\to\mathbb{R}

is differentiable on

(0,1)

and

f(0)=1

f(1)=0

. Then, there exists

0<x_0<1

such that

|f'(x_0)| \geq 2018 f(x_0)^{2018}