SE-SKOLORNAS 1963 Problem 6

The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$ , $|f''(x)| \le B$ for some positive $A, B$ (and all $x$ ). Show that $|f'(x)| \le C$ , for some fixed $C$ , which depends only on $A$ and $B$ . What is the smallest possible value of $C$ ?