= Solution
The function
$$
f(\mathcal M)=\mathcal M^{3/2}+3\mathcal M^{-1/2}
$$
has its unique minimum at $\mathcal M=1$, where $f(1)=4$. At large $r$, mass conservation and the reservoir boundary conditions give $u\propto r^{-2}$ and hence $\mathcal M\to0$. The physical solution therefore begins on the subsonic branch $0<\mathcal M<1$. It cannot pass smoothly to $\mathcal M>1$ without reaching the minimum, where the two algebraic branches meet. For $\gamma=5/3$ that meeting can occur only in the limiting central behavior of the critical solution, so the flow remains subsonic for every $r>0$.
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