Let $f$ be a non-negative $C^2$ function on a compact domain $\Omega$ in $\mathbb R^n$. I am trying to prove the inequality
$\|\sqrt f\|_{C^{0,1}(\Omega)}\leq C(1 + \|f\|_{C^{1,1}(\Omega)})$
where $C^{k,\alpha}(\Omega)$ denotes the Holder space. It seems like this should be a consequence of the mean value theorem and/or the fundamental theorem of calculus, but I am not seeing an elementary proof of this nature. Any suggestions?