Question 154
Let $\mathcal{C}$ be the set of functions $f:[0,1] \rightarrow \mathbb{R}$, twice differentiable on $[0,1]$, which have at least two zeros, not necessarily distinct, in $[0,1]$ and satisfy $\left|f''(x)\right| \leq 1$ for every $x$ in $[0,1]$. Determine the maximum value that the integral
$$
\int_{0}^{1}|f(x)| \, \mathrm{d} x
$$
can take when $f$ ranges over the set $\mathcal{C}$, and the functions that attain this maximum.
(A differentiable function $f$ has two zeros at the same point $a$ if $f(a)=f'(a)=0$.)
$$
\int_{0}^{1}|f(x)| \, \mathrm{d} x
$$
can take when $f$ ranges over the set $\mathcal{C}$, and the functions that attain this maximum.
(A differentiable function $f$ has two zeros at the same point $a$ if $f(a)=f'(a)=0$.)