Finding optimum market value of vintage car
Note that, from your SOC:
\begin{align}
P''(t) &amp;= V''(t)e^{-rt} -  rV'(t)e^{-rt}-  rV'(t)e^{-rt} + r^2V(t)e^{-rt} + rme^{-rt}\\
&amp;=e^{-rt}\Bigl[\underbrace{\color{red}{V''(t)-rV'(t)}}_{=D}-r\underbrace{\bigl[V'(t)-rV(t)-m\bigr]}_{=0\text{ from FOC}}\Bigr]
\end{align}