jmc

Plugging $x=2$ into the expression for $f(x)$, we find $f(2)=c(2^3)-9(2)+3=8c-18+3=8c-15$. Since we know that $f(2)=9$, $$\begin{array}{rl}f(2)& =9\\ \Rightarrow 8c-15& =9\\ \Rightarrow 8c& =24\\ \Rightarrow c& =\boxed{3}\end{array}$$