I have this function $163.6718+48.7542n-1.376957n^2+0.01518444n^3$. And want to find the big-O constant as demonstrated here for n³. Except for n³, I have nlog(n) as an upper-bounding function.
The function does not have to hold true for n>50
The (in)equation would look like
$163.6718+48.7542n-1.376957n^2+0.01518444n^3 <= c*nlog(n)$
I do not know how to get divide by log(n) or if it is even possible. Anyway, I am trying to find $c$ here.
No such $c$ exist, let $n=1$,
Then the left hand side is positive but the right hand side is $0$.
Suppose you want to make the statement to be true for $n \in \{2, \ldots, 50\}$.
Let $f(n)$ be your cubic polynomial, you just have to evaluate
$\frac{f(n)}{n \log(n)}$ for these $49$ values and pick the maximum value.