SE-SKOLORNAS 1976 Problem 5

$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0)=1$ , $f'(0)=0$ and $(1+f(x))f''(x)=1+x$ . Show that $f$ is increasing and that $f(1) \leq 4/3$ .