MathNet 2005 — Part 01 Question 18

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a smooth function such that $f'(x)^2 = f(x) f''(x)$ for all $x$. Suppose $f(0) = 1$ and $f^{(4)}(0) = 9$. Find all possible values of $f'(0)$.