Assume that $f$ is twice diffentiable on $R$,and such
$f(x)$and $f'(x)$ are bounded on $R$
and following I can’t any work,Thank you very much!
By definition, $g$ is non-negative and differentiable; moreover,
$$g'(x)=f'(x)\cdot\left(2f(x)+f”(x)\right)=-x\cdot (f'(x))^2,\quad\forall x\in \Bbb R.$$
Therefore, $g$ is increasing on $(-\infty,0]$ and decreasing on $[0,+\infty)$, so $g(\Bbb R)\subset [0, g(0)]$. The conclusion follows.