$L^1$ bound and derivative bound implies $L^p$ bound

Question

Suppose non-negative $f: [0, \infty)\to\mathbb{R}_{\geqslant 0}$ is differentiable, with $f(0)=0$ and $f'\leqslant 1/2$. Show that if $\int_0^\infty f\,dx<\infty$, then for all $p>1$ we have $$ \int_0^\infty f^p\,dx \leqslant \left(\int_0^\infty f\,dx\right)^{(p+1)/2}. $$