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A ship must travel between two ports, A and B, which is a total distance of D. The cost per-hour of operating the ship is c = w + vn, where w > 0 represents the hourly wages of the ship's employees, v is the ship's velocity, and n ≥ 2 is a parameter. The objective is to choose v so that the trip is made at miminum cost. (Hint: velocity is equal to distance divided by the time taken on the trip, or what is the same, time taken is equal to distance divided by velocity)
1. What is the optimal v, and how does it change with D and with w?
2. If n = 2, is the optimal velocity concave or convex in w?
3. Is the optimal velocity concave or convex in w when n > 2?
4. Consider the shape of optimal velocity in w (i.e. concave or convex) for the two concrete cases of n = 3 and n = 4.
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