= Solution
Write $r=r_0+\eta$ and keep $L$ fixed. The <small radial oscillations in a power-law potential> have squared <angular frequency>
$$
\omega_r^2=\frac{V_{\mathrm{eff}}''(r_0)}m
=\frac{3L^2}{m^2r_0^4}+\frac{\beta k(k-1)}m r_0^{k-2}
=\frac{\beta k(k+2)}m r_0^{k-2}.
$$
The circular orbital <angular frequency> has magnitude $\Omega=|L|/(mr_0^2)$, so $\Omega^2=(\beta k/m)r_0^{k-2}$. Hence
$$
\boxed{\omega_r=\Omega\sqrt{k+2},\qquad\frac{\omega_r}{\Omega}=\sqrt{k+2},\qquad
f_r=\frac{\Omega}{2\pi}\sqrt{k+2}.}
$$
A restoring radial oscillation exists for $k>-2$, subject to the attractive-force condition in part (b). For $k<-2$ the radial stationary point is unstable; $k=-2$ is the degenerate boundary. This agrees with <circular-orbit stability for a power-law central force>, since the actual force is proportional to $-r^{k-1}$.
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