= Solution
At a finite smooth <sonic point> $r_*$, the equation from (a) requires
$$
u_*^2=c_*^2,\qquad
c_*^2=\frac{A\beta}{2r_*^\beta}.
$$
Set $\delta=\gamma-1$ and $q=2/\beta-1/2$. Substitution into the <Bernoulli function> gives
$$
\mathcal B=c_*^2\left(\frac1\delta-q\right).
$$
An inflow from a warm reservoir at infinity has $\mathcal B=c_\infty^2/\delta>0$; an outflow reaching infinity has $\mathcal B=u_\infty^2/2+c_\infty^2/\delta\geq0$. Therefore a nondegenerate <transonic spherical flow in a power-law potential> requires
$$
\boxed{2\beta-(4-\beta)(\gamma-1)>0.}
$$
For the usual range $0<\beta<4$, the <critical adiabatic exponent for spherical power-law flow> is consequently
$$
\boxed{f(\beta)=\frac{4+\beta}{4-\beta},\qquad 1<\gamma<f(\beta).}
$$
The printed assumption $\beta>0$ also permits $\beta\geq4$. In that range the boxed inequality holds for every finite $\gamma>1$: there is no finite upper bound. Equivalently, take $f(\beta)=+\infty$ for $\beta\geq4$. Extending the rational expression beyond $\beta=4$ would give an incorrect restriction.
The strict inequality also follows from local <sonic-point slope discriminant> analysis, including the potentially cold zero-energy endpoint. Put $X=(r_*/u_*)u_*'$. Since <mass conservation> implies $(c^2)'/c^2=-\delta(u'/u+2/r)$, differentiating the <sonic point> equation at $r_*$ yields
$$
(\delta+2)X^2+4\delta X+4\delta-2\beta=0.
$$
Its discriminant is $8[2\beta-(4-\beta)\delta]$. Moreover, the logarithmic <Mach number> derivative at the <sonic point> is
$$
\left.\frac{d\ln(|u|/c)}{d\ln r}\right|_*
=\frac{\delta+2}{2}X+\delta
=\pm\sqrt{\frac{2\beta-(4-\beta)\delta}{2}}.
$$
A genuine crossing has two distinct branches and a nonzero derivative. At equality the crossing degenerates. In particular, for $\gamma=f(\beta)$ and $\mathcal B=0$, <mass conservation> and the <polytropic equation of state> make the <Mach number> constant along the scale-invariant solution; a solution that is sonic there is sonic everywhere, rather than crossing an isolated <sonic point>.
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