Can anyone help me with this problem? There is already a post explaining it but I could really use further explanation. The problem states:

___________

Let C(R) be the collection of all continuous functions from R to R (Set of reals to set of reals). Then C(R) is a vector space with pointwise addition and scalar multiplication defined by.

(f+g)(x)=f(x)+g(x) and (rf)(x)=rf(x).

for all f, g in C(R) and all r, x in R. Which of the following are subspaces of C(R)?

I: {f: f is twice differentiable and f''(x)-2f '(x)+3f(x)=0 for all x}

II: {g: g is twice differentiable and g''(x)=3g'(x) for all x}

III:{h: h is twice differentiable and h''(x)=h(x)+1 for all x}

A) I only

B)I and II only

C)I and III only

D) II and III only

E) I, II, and II

The answer is suppose to be B.

_______________

First I tried to find a function f that would satisfy the condition stated in I. I chose the function f(x)=e^(sx), for some unknown s, as a possible solution to the differential equation described in I. However, as I try to solve for the unknown constant I end up with an imaginary number which would imply that f is not within the set C(R).

Work to find the unknown constant in the proposed function:

f''(x)-2f'(x)+3f(x)=0, assume f(x)=e^(sx) then

s^2*e^(sx)-2s*e^(sx)+3*e^(sx)=0 ---->

s^2-2s+3=0.

The solution to s is s=1-i(2)^(1/2) which is an imaginary number. This would also make f(x) imaginary and therefor not a part of C(R).

I know I'm really lost on the problem and probably nowhere on the right track. Anyone that could break this problem down for me would receive my undying gratitude.