Algorithm Analysis: How to calculate coefficients of asymptotic notation?

An algorithm with running time $f(n)$ is said to be $\theta(g(n))$ if the following holds:

$c_1g(n) \leq f(n) \leq c_2g(n)$ for all $n>n_0$ and some value of $c_1$ and $c_2$.

For example, $f(n)=\frac{n^2}{2}-3n$ is $\theta(n^n)$.

To prove this, we need to find at least one set of values for $n_0, c_1$ and $c_2$ that satisfies above inequality. Below is a simple procedure to do that:

1. To calculate $c_1$, obtain a value of n for which $f(n)$ is positive and solve left side inequality with that value of $n$.
• $n_0=7$ and $c_1\leq1/14$
2.  To calculate $c_2$, just put $n=1$ and solve the right side inequality.
• $n_0=1$ and $c_2=1/2$

The basic idea is to make sure that $c_1$ and $c_2$ are positive.