Need help for 2 interesting questions - rolle's and continuity

1. Let f and g be functions such that f'' and g'' exist everywhere on R. For a < b, suppose that f(a)=f(b)=g(a)=g(b)=0, and $\displaystyle g''(x) \not= 0$ for every x $\displaystyle \in$ (a,b).

(i) Prove that $\displaystyle g(x) \not= 0$ for every x $\displaystyle \in$ (a,b). *I know how to do this, if I'm not wrong, use Rolle's Theorem a few times.*

(ii) **Show that there exists a number c $\displaystyle \in$ (a,b) for which **$\displaystyle \frac{f(c)}{g(c)} = \frac{f''(c)}{g''(c)}$ *No idea. Does it have something to do with the Mean Value Theorem?*

2. Let f be a continuous function on R such that $\displaystyle \lim_{x\to0}\frac{f(x)}{x}$ exists. Define the function

g(x) = $\displaystyle \int_0^1 f(xt) dt, x \in R$

**Determine whether g' is continuous at x = 0. Justify your answer.**