Determine if $ f(x) = 4x^2+x+1$ is $O(x^2)$. If so find the constants $C$ and $k$ to verify that the function is $O(x^2)$
My solution is:
\begin{align} & |f(x)| \le C|x^2| \ \ \ \ \ \ \forall x > k\\[8pt] &|4x^2+x+1| \le C(x^2) \ \forall x > 1 \\[8pt] & 4x^2+x+1 \le 4x^2 +x^2+x^2 \ \ \ \forall x > 1\\[8pt] & 4x^2+x+1 \le 6(x^2)\\[8pt] & k=1 \text{ and } C=6 \\[8pt] & f(x) = O(x^2) \end{align}
Is this correct? Im unsure of my methodology.
That Inequality is certainly true and your methodology seems correct, however I am curious as to why the question asks you to find "the constants $C$ and $k$," as opposed to just "constants $C$ and $k$." For example, a similar methodology could yield $C$ = 5 and $k$ = 2. $$ \begin{align} & |f(x)| \le C|x^2| \ \ \ \ \ \ \forall x > k\\[8pt] &|4x^2+x+1| \le C(x^2) \ \forall x > 2 \\[8pt] & 4x^2+x+1 \le 4x^2 + x^2 \ \ \ \forall x > 2\\[8pt] & 4x^2+x+1 \le 5(x^2)\\[8pt] & k=2 \text{ and } C=5 \\[8pt] \end{align}$$