Need some help figuring out how to solve the following problems

42. What is the greatest value of b for which any real value function f that satisfies the following properties must also satisfy f(1)<5?

(i) f is infinitely differentiable on the real numbers

(ii) f(0)=1, f'(0)=1, and f''(0)=2; and

(iii) |f'''(x)|<b for all x in [0,1]

a. 1

b. 2

c. 6

d. 12

e. 24

64. Let V be the real vector space of all real-valued functions defined on the real numbers and having derivatives of all orders. If D is the mapping from V into V that maps every function in V to its derivative, what are all the eigenvectors of D?

a. All nonzero functions in V

b. All nonzero constant functions in V

c. All nonzero functions of the form ke^(lk), where k and l are real numbers

d. All nonzero functions of the form sum from i=0 to k of c sub i times x^i where where k>0 and csub i's are real numbers

e. There are no eigenvectors of D

the answers are d and c any help is greatly appreciated