Question: When a fish swims upstream at a speed v against a constant current \({{v}_{w}}\), the energy it expends in traveling to a point upstream is given by a function of the form \(E(v)=\frac{C{{v}^{k}}}{v-{{v}_{w}}}\) where C > 0 and k > 2 is a number that depends on the species of fish involved.

a. Show that E(v) has exactly one critical number. Does it correspond to a relative maximum or a relative minimum?

b. Note that the critical number in part (a) depends on k. Let F(k) be the critical number. What can be said about F(k) if k is very large?