Differential Equations for Exponential Growth and Decay

Let $f$ be a real smooth (i.e. infinitely-differentiable) function such that $f'(x)^2 = f(x) f''(x)$ for all real $x$. Suppose $f(0) = 1$ and $f^{(4)}(0) = 9$. How many possible values of $f'(0)$ are there?

Source: HMMT February 2005