I'm really struggeling with this obviously easy Lagrange question. Been at it for two days and don't really get the hang of where to start.. can someone pointme in the right direction?
A consortium, having the monopoly in a route, wishes to maximise their earnings. Assume a single type of cargo, its annual quantity transported by the consortium (variable $X$) linearly changing with the freight rate (variable $r$) charged (for $X = 10$ million tonnes, $r = 0$ and for $X = 0$, $r =$ $20/tonne).
The fixed and variable cost in this route ($FC$ and $VC$, respectively) are considered to be functions of two variables, $S$ (carrying capacity committed by the consortium on the route) and $X$ (quantity of cargo transported by the consortium). Assume that $FC$ and $VC$ are ($S$ and $X$ in million tonnes per year, $FC$ and $VC$ in dollars per year): $$ FC = 1000000 \times S $$ $$ VC= \begin{cases} 500000\left( \log S - \log (S-X) \right) &\hbox{if $X<S$}\\ +\infty &\hbox{otherwise} \end{cases} $$
a. Assuming a carrying capacity committed by the consortium on the route of 7.5 million tonnes per year, determine the freight rate that would maximise earnings.
The earning is $RV-FC-VC$, where $RV$ is the revenue.
You can find revenue by $X\cdot r$ where $X$ is represented linearly with respect to $r$ using the given conditions.
So you will end up with an equation with respect to $r$. I believe you can find the extreme values using first derivative or second derivative test.
I hope this can give you a start.