Senterville is the headquarters of Greedy service to two nearby towns, Cablevision Inc. The cable company is about to expand Centerville to both Springfield and Shelbyville. There needs to be a cable connection between the three towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuration. Centerville is located at (10,0) in the xy-plane; Springfield is at (0, 5), and Shelbyville is at (0, 0). The cable runs from Centerville to some point (0, T) on the z-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (0, T) that will minimize the amount of cable needed. Justify your answer. To solve this problem, we need to minimize the following function of T: f(T). We find that f(T) has a critical number at T = 0. To verify that f(T) has a minimum at this critical number, we compute the second derivative f''(T). Thus, the minimum length of cable needed is a positive number.