Maximizing Integral With Bounded Second Derivative

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 all $x$ in $[0,1]$. Determine the maximum possible value of the integral $$ \int_{0}^{1}|f(x)| \, \mathrm{d} x $$ as $f$ ranges over the set $\mathcal{C}$, and the functions that achieve this maximum. (A differentiable function $f$ has two zeros at the same point $a$ if $f(a)=f'(a)=0$.)