Solution
= Solution
In <big O notation>, $f(n)=O(g(n))$ means that there exist $C>0$ and $n_0$ such that $|f(n)|\le C|g(n)|$ for every integer $n\ge n_0$. For the required <polynomial versus exponential growth> comparison, $3^3=3^3$. If $n^3\le3^n$ and $n\ge3$, then
$$
(n+1)^3=n^3\left(1+\frac1n\right)^3\le3^n\left(\frac43\right)^3<3^{n+1}.
$$
By <mathematical induction>, $n^3\le3^n$ for all $n\ge3$. Thus \b[$n^3=O(3^n)$], with $C=1$ and $n_0=3$.