Assume that the cost of the petrol burnt (per hour) in driving a motor boat varies as the cube of its velocity. Then the most economical speed of the boat when going against a current of 6km per hour is -------

Let p be the petrol burnt and v be the velocity of the motor boat 
$\therefore {\mathrm{p\alpha v}}^3\Rightarrow \mathrm{k}\cdot {\mathrm{v}}^3$ Actual speed of the boat = v - 6 
Let ‘s’ be the distance travelled by the boat in time $\mathrm{t}\Rightarrow \mathrm{t}=\frac{\mathrm{s}}{\mathrm{v}-6}$
Cost of the fuel $=\frac{\mathrm{s}}{\mathrm{v}-6}{\mathrm{kv}}^3=\mathrm{f}\left(\mathrm{v}\right)$
$\begin{array}{l}{\mathrm{f}}^1\left(\mathrm{v}\right)=\mathrm{k}\cdot \mathrm{s}\frac{2{\mathrm{v}}^3-18{\mathrm{v}}^2}{(\mathrm{v}-6{)}^2};{\mathrm{f}}^1\left(\mathrm{v}\right)=0\\ 2{\mathrm{v}}^3-18{\mathrm{v}}^2=0;\mathrm{v}-9=0;\mathrm{v}=9\Rightarrow {\mathrm{f}}^{11}\left(\mathrm{v}\right)>0\end{array}$