Exercise Let $f\colon \mathbb{R} \to \mathbb{R}$ be $C^2$ and nonnegative. Prove that
$\big( f'(x)\big)^2 \le 2f(x) \lVert f''\rVert_{\infty}.$
I have found this innocent-looking little exercise... but I must admit I'm stuck on it. I have tried various roads, the most promising of them being integration by parts: (Notation: $\Delta_hf(x)=(f(x+h)-f(x))/h$)
$\frac{1}{h}\int_x^{x+h} (f'(t))^2\, dt = f(x+h)\Delta_hf'(x)+f'(x)\Delta_hf(x)-\frac{1}{h}\int_{x}^{x+h}f(t)f''(t)\, dt;$
I had hoped to bound this identity from above then have $h \to 0^+$. But I got nowhere.
Other approaches used Taylor expansions. All I got with those was a weaker estimate (that I reported here).
Can somebody give me a hint? I can't stop thinking at this exercise but I've got some work to do! :-)