 Solution

Set ${\theta }_{\mathrm{rot}}=hc\tilde{B}/k_{\mathrm{B}}$, the rotational temperature. For a continuous nonnegative variable $J$,Equating this derivative to zero gives the most populated rotational levelThe second derivative is $-4/(2J+1{)}^2-2{\theta }_{\mathrm{rot}}/T<0$, so this is a maximum when $J_{\ast }\ge 0$. At lower temperatures the constrained continuous maximum is $J=0$. For the physical integer maximum compare the nearby allowed levels explicitly.