Answer
The minimum dissipation rate is $~~0$
Work Step by Step
Note that $~~I_{rms} = \frac{\mathscr{E}_{rms}}{Z}$
The average rate at which energy is dissipated in the resistor is a minimum when $I_{rms}$ is a minimum.
This occurs when we maximize the impedance $Z$
$Z = \sqrt{R^2+(X_L-X_C)^2}$
$Z = \sqrt{R^2+(\omega_d~L-1/\omega_d~C)^2}$
Note that when $C = 0$, then $Z \to \infty$
In this case, $I_{rms} = 0$
We can find $P_{ave}$:
$P_{ave} = I_{rms}^2~R = (0)^2~R = 0$
The minimum dissipation rate is $~~0$
