 Solution

For $A,B>0$, differentiating the simplified Köhler theory curve givesThe derivative is positive below $r_c$ and negative above it, so $r_c$ is the Köhler activation radius, the unique maximum. At that point $B/r_c^3=A/(3r_c)$, givingThe curve crosses $S=1$ at $r=\sqrt{B/A}=r_c/\sqrt{3}$, reaches $S_c$ at $r_c$, and approaches $1$ from above as $r\to \infty$. Its formal divergence to negative values as $r\to 0$ is outside the dilute-chemical solution approximation; a finite dry solute core and composition-dependent properties must replace that extrapolation.