Big O notation checking if C,K can be used as witnesses I have an assignment where I have to find out if $(C,k) = (6,1)$ can be used as witnesses for $$f(x) = 3x^3 + 2x + 4\ \text{ is }\ O(x^3).$$
Edit: I'm looking for a general answer
 A: Once we "unpack" the definitions, what this means is that you want to check if $$3x^3 + 2x + 4 \le 6x^3 \qquad \text{ for all } x\ge 1.$$
(Actually, both sides of the inequality should have absolute values around them, but we can drop those here as everything will be positive.)
In this particular case, it's straightforward to check that the inequality is false when you set $x=1$, for example. 
On the other hand, we have $2x \le 2x^3$ and $4 \le 4x^3$ for all $x \ge 1$, so $$3x^3 + 2x + 4 \le 3x^3 + 2x^3 + 4x^3 = 9x^3 \qquad \text{ for all }x \ge 1$$ and the pair $(C,k) = (9,1)$ can be used as witnesses.
But the inequality work is honestly a much less important skill for you to learn here than the skill of applying definitions. You need to get to the point where you can automatically translate the statement

$(C,k)$ can be used as witnesses for the statement $f(x) = 3x^3 + 2x+4$ is $O(x^3)$

into the statement 

$|3x^3+2x+4| \le C|x^3|$ for all $x \ge k$

as soon as you learn the definition of witnesses for a $f(x) \in O(g(x))$ statement, because this skill is important with all definitions you learn.
