How to derive 2nd order Butterworth condition for the damping coefficient mathematically?
You have to check the next derivative, because the first one is zero anyway at $\omega=0$. The second derivative is
$$3\omega^2+2\zeta^2-1\Big|_{\omega=0}=2\zeta^2-1\tag{1}$$
which can be made zero by choosing $\zeta=1/\sqrt{2}$.