 Solution

A product state factors as $|{\psi }_1⟩\otimes |{\psi }_2⟩$; an entangled state admits no such factorization. The state $|J,J⟩$ is a product state. When $j_1,j_2>0$, each state with $m=J-1$ displayed in part (iii) has two nonzero terms with orthogonal states in both subsystems, so its Schmidt rank is two and it is entangled. If one subsystem has spin zero, only the $j=J$ multiplet remains and its states factor through the unique state of that subsystem.