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 for every x (a,b).

(i) Prove that for every x (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 (a,b) for which ** *No idea. Does it have something to do with the Mean Value Theorem?*

2. Let f be a continuous function on R such that exists. Define the function

g(x) =

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