Answer
$L=C$
Work Step by Step
$v$ = $K\sqrt {\frac{L}{C}+\frac{C}{L}}$
$\frac{dv}{dL}$ = $\frac{K}{2\sqrt {\frac{L}{C}+\frac{C}{L}}}\left(\frac{1}{C}-\frac{C}{L^{2}}\right)$ = $0$
$\frac{1}{C}$ = $\frac{C}{L^{2}}$
$L$ = $C$
This gives the minimum velocity since
$v'$ $\lt$ $0$ for $0$ $\lt$ $L$ $\lt$ $C$
$v'$ $\gt$ $0$ for $L$ $\gt$ $C$
