IMC 1997 Problem 7

Let $f\in C^3(\mathbb{R})$ nonnegative function with $f(0)=f'(0)=0, f''(0)>0$ . Define $g(x)$ as follows: $$\{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 g(x)=0 &\text{for}& x=0\end{array}$$

(a) Show that $g$ is bounded in some neighbourhood of $0$ . (b) Is the above true for $f\in C^2(\mathbb{R})$ ?