AHSME 1975 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 are true? $I.\ f(0) = 1 \ \ II.\ f(-a) = \frac{1}{f(a)}\ \text{for all}\ a \\ III.\ f(a) = \sqrt[3]{f(3a)}\ \text{for all}\ a IV.\ f(b) > f(a)\ \text{if}\ b > a$ $\textbf{(A)}\ \text{III and IV only} \textbf{(B)}\ \text{I, III, and IV only} \\ \textbf{(C)}\ \text{I, II, and IV only} \textbf{(D)}\ \text{I, II, and III only} \textbf{(E)}\ \text{All are true.}$