Proving that for $f\in C^2(a,\infty)$ then $\max_{(a,\infty)} |f'(x) |\le 4\max_{(a,\infty)}|f(x)|\max_{(a,\infty)} |f''(x) |$.

Question

Let $f\in C^2(a,\infty)$ and $M_k=\max_{(a,\infty)} |f^{(k)}(x) |$, $k=0,1,2$, prove that $$M_1^2\le 4M_0M_2.$$

Firstly, I know that if $f, g\in C^2(a,\infty)$ so is $fg\in C^2(a,\infty)$. Any hint on how to proceed is highly appreciated. So that I would know how to go about proving this if I am to use the given condition.