Second-Order Linear Equations · Challenge problem

A $2$ kg mass on a spring of constant $18$ N/m moves by $2y'' + cy' + 18y = 0$ with adjustable damping $c > 0$. Every solution decays at least as fast as $e^{-\sigma t}$ for some rate $\sigma(c)$, the smallest of the decay rates of the characteristic roots. What is the largest possible value of $\sigma(c)$ over all $c > 0$?