Textbook Problem

For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to $v^3$. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current $u\left(u<v\right)$, then the time required to swim a distance L is $L/\left(v-u\right)$ and the total energy E required to swim the distance is given by

where a is the proportionality constant.

(a) Determine the value of v that minimizes E.

(b) Sketch the graph of E.

Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed.