By conservation of mechanical energy:
$$
E(r_0)=-\frac{GMm}{r_0}+\frac{1}{2}\mu \left (\dot{r_0}^2+r_0^2 \omega_0^2 \right)
$$
where R0 =Rmax. Because our body is located at the apoapsis the radial velocity is 0. Hence:
$$
E(r_0)=-\frac{GMm}{r_0}+\frac{1}{2}\mu (r_0\omega_0)^2
$$
By the...