Modern Algebra help! Let F be the additive group of functions mapping R-R with derivatives of all orders. ?

You are looking for all functions that phi maps to 0, that is ker(phi). In other words you are looking for all functions those third derivative is zero. So here is a hint: polynomials have the property that if you differentiate them enough times you get zero, but if the the degree is too high maybe you have to differentiate more than 3 times.