as mass is constant \(\Rightarrow m =\rho V =\) constant

\(V = constant\)

\(\pi^{2} R l=\) constant \(\Rightarrow R ^{2} L =\) constant

\(2 RL + R ^{2} \frac{ dL }{ dR }=0\)\(...(2)\)

From equation \((1)\)

\(\frac{ dI }{ clR }= M \left(\frac{2 R }{4}+\frac{2 L }{12} \times \frac{ dI }{ dr }\right)=0\)

\(\frac{ R }{2}+\frac{ L }{6} \frac{ dI }{ dR }=0\)

Substituting value of \(\frac{ dI }{ d R }\) from eqution \((2)\)

\(\frac{ R }{2}+\frac{ L }{6}\left(\frac{-2 L }{ R }\right)=0\)

\(\frac{ R }{2}=\frac{ L^{2}}{3 R } \Rightarrow \frac{ L }{ R }=\sqrt{\frac{3}{2}}\)