This question is mostly about understanding definitions.

For (a) prove that G satisfies all the group axioms under addition. Let f, g, h be infinitely differentiable at every point x in R

Closure : f + g is infinitely differentiable at every point x in R

Associativity : (f + g) + h = f + (g + h)

Identity : Find a function e in G such that for all functions f in G such that f + e = e + f = f

Inverse : For all f in G find an inverse ~f in G such that f + ~f = e

Is G a group under multiplication? Well, is f^{-1} always infinitely differentiable when f is infinitely differentiable? Try f(x) = x

For (b) and (c)

The kernel of a homomophism are all elements of the domain group that map to the zero element, e, of the codomain.

What is the zero element of the codomain? This is e you found for G in the group axioms. In this problem e is the zero function zero(x) = 0.

So for (b), the problem is asking for which functions f in G does phi(f) = zero, i.e. which functions has f'(x) = 0 for all x in R?

Similar for (c), the problem is asking for which functions f in G does psi(f) = zero, i.e. which functions has f''(x) - f(x) = 0 for all x in R?