1975 AHSME Problem 21

Suppose f(x) is defined for all real numbers x; f(x)>0 for all x, and f(a)f(b)=f(a+b) for all a and b. Which of the following statements is true?

\text{I. } f(0)=1

\text{II. } f(-a)=1/f(a) \text{ for all } a

\text{III. } f(a) = \sqrt[3]{f(3a)} \text{ for all } a

\text{IV. } f(b)>f(a) \text{ if } b>a

\textbf{(A)}\ \text{III and IV only} \qquad \textbf{(B)}\ \text{I, III, and IV only} \qquad \textbf{(C)}\ \text{I, II, and IV only} \qquad \textbf{(D)}\ \text{I, II, and III only} \qquad \textbf{(E)}\ \text{All are true.}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.