Question 29
Let $C$ be the set of all functions $f: [0, 1] \to \mathbb{R}$, twice differentiable on $[0, 1]$, with at least two (not necessarily distinct) zeroes in $[0, 1]$ and such that $|f''(x)| \le 1$, for all $x$ in $[0, 1]$. Find the maximum possible value of the integral
$$
\int_0^1 |f(x)| \, dx,
$$
when $f$ runs through $C$, and find the functions for which the maximum is reached. (A double zero is a point $a$ such that $f(a) = f'(a) = 0$.)
$$
\int_0^1 |f(x)| \, dx,
$$
when $f$ runs through $C$, and find the functions for which the maximum is reached. (A double zero is a point $a$ such that $f(a) = f'(a) = 0$.)