Finding optimum market value of vintage car I am trying to find first and second order conditions for the following problem:
$$P(t) = V(t)e^{-rt}- \int^t_0 me^{-r\tau}d \tau $$
I managed to find the first order condition:
$$P'(t) = V'(t)e^{-rt} -  rV(t)e^{-rt} - me^{-rt} =0 \\
V'(t) = rV(t)+m $$
According to our textbook this is correct, but I have problem finding SOC.  We learned before that SOC are found by taking second order derivative and examining if it is &gt; or &lt; than 0.
When I try to find $P''(t)$ I get:
$$P''(t) = V''(t)e^{-rt} -  rV'(t)e^{-rt}-  rV'(t)e^{-rt} + r^2V(t)e^{-rt} + rme^{-rt}$$
But the textbook says that the second order condition should be:
$$D=V''(t) -rV'(t)$$
this looks like the derivative of $V'(t) = rV(t)+m \implies V'(t) - rV(t) - m = 0 $
but I don't understand why that is correct. Previously when deriving SOC we always just took the 2nd derivative of the function so how come that does not work here?
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}