Calculate the absolute value of the complex charge in the RLC circuit:
$$Q(t)=\frac{V_0e^{i\omega t}}{-\omega^2L+i\omega R+\frac{1}{c}}.$$
Find the frequency where $|Q(t)|$ is maximum.
This question is from Mathematical methods, by Boas. I am not so sure what the question is asking?
If I take absolute value then I will have:
$$\left|Q(t)\right|=\left|\frac{V_0e^{i\omega t}}{-\omega^2L+i\omega R+\frac{1}{C}}\right|= \frac{\left|V_0e^{i\omega t}\right|}{\left|-\omega^2L+i\omega R+\frac{1}{C}\right|}=\frac{V_0}{\left|-\omega^2L+i\omega R+\frac{1}{C}\right|}.$$
You're on the right track. First, evaluate the $|.|$ in the denominator to get an expression in terms of $R,L,C$, and $\omega$:
$$ \left| \left(\frac{1}{C}-L\omega^2\right)+i\left(R\omega \right)\right| = \sqrt{\left(\frac{1}{C}-L\omega^2\right)^2 + \left(R\omega\right)^2} $$
Now differentiate $|Q(t)|$ w.r.t. $\omega$, set it to $0$ and solve for $\omega$. This will be the frequency at which $|Q(t)|$ is maximum.
PS: If you want to be rigorous, differentiate $|Q(t)|$ again and use the second derivative to show that the solution is indeed a maximum, not a minimum or a saddle point.