The fuel charges for running a train are proportional to the square of the speed generated in  $km/ h$ and the cost is Rs 48, at 16 km/h. If the fixed charges amount to Rs.$300/h,$ the most economical speed is

Let the speed of the Train be v and distance to be covered be so that total time taken is s/v hours. Cost of fuel per hour = kv² (k is constant) Also 48 = k16² by given condition
$\therefore \mathrm{k}=\frac{3}{16}$
$\therefore$Cost of fuel per hour = $=\frac{3}{16}{\mathrm{v}}^2$
Other charges per hour are 300
Charges per hour are $\frac{3{\mathrm{v}}^2}{16}$+300
Total expenses for the journey
$\begin{array}{l}\mathrm{E}=\left(\frac{3{\mathrm{v}}^2}{16}+300\right)\frac{\mathrm{s}}{\mathrm{V}}\\ \frac{\mathrm{dE}}{\mathrm{dv}}=\mathrm{S}\left(\frac{3}{16}-\frac{300}{{\mathrm{v}}^2}\right)\\ \frac{\mathrm{dE}}{\mathrm{dv}}=0\\ \mathrm{V}=40;\frac{{\mathrm{d}}^2\mathrm{E}}{{\mathrm{dv}}^2}>0\end{array}$
And hence E is minimum 
Hence to most economical speed is 40 km/h.