Putnam 1989 B3

Let $f$ be a function on $[0,\infty)$, differentiable and satisfying $$f'(x)=-3f(x)+6f(2x)$$ for $x>0$. Assume that $|f(x)|\le e^{-\sqrt{x}}$ for $x\ge 0$ (so that $f(x)$ tends rapidly to $0$ as $x$ increases). For $n$ a non-negative integer, define $$\mu_n=\int_0^\infty x^n f(x) dx$$ (sometimes called the $n$th moment of $f$).

a) Express $\mu_n$ in terms of $\mu_0$.

b) Prove that the sequence $\{\mu_n \frac{3^n}{n!}\}$ always converges, and that the limit is $0$ only if $\mu_0=0$.