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

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. Sketch the graph of F(k). What can be said about F(k) if k is very large?

E(v) = Cut V - Vw