MathDB

Let

\mathcal{C}

be the set of all twice differentiable functions

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

with at least two (not necessarily distinct) zeros and

|f''(x)| \le 1,

for all

x \in [0,1].

Find the greatest value of the integral

\int\limits_0^1 |f(x)| \mathrm{d}x

when

f

runs through the set

\mathcal{C},

as well as the functions that achieve this maximum.
Note: A differentiable function

f

has two zeros in the same point

a

f(a)=f'(a)=0.

Problem Statement