What needs to be done in part (a) is to check for which values of a and b does the function f have a limit for x tending to 0. If f does have a (finite) limit, then you can extend f to all of R by letting f(0) be that limit.
As for part b and c, you do something very similar. Well, first check if the function f is differentiable in the first place. You know that if f is to be extended to a differentiable function, then the value of f at 0 must be the limit of f(x) for x-->0. Check which choices of a and b make the extension differentiable at 0. Similarly for the second derivative.