Curve below two lines from a bounded derivative

Let $f(x)$ be a nonnegative differentiable function on $[a,b]\subset\mathbb R$ such that $f(a)=0=f(b)$ and $|f'(x)|\le4$. Let $L_1$ and $L_2$ be the straight lines given by the equations $y=4(x-a)$ and $y=-4(x-b)$, respectively. Then which of the following statements is (are) TRUE?