Centerville is the headquarters of Greedy Cablevision Inc. Thecable company is about to expand service to two nearby towns,Springfield and Shelbyville. There needs to be cable connectingCenterville to both towns. The idea is to save on the cost of cableby arranging the cable in a Y-shaped configuation. Centerville islocated at (12,0) in the xy-plane, Springfield is at (0,9), andShelbyville is at (0,?9). The cable runs from Centerville to somepoint (x,0) on the x-axis where it splits into two branches goingto Springfield and Shelbyville. Find the location (x,0) that willminimize the amount of cable between the 3 towns and compute theamount of cable needed. Justify your answer.
To solve this problem we need to minimize the following functionof
We find that f(x) has a critical number at x=
To verify that f(x) has a minimum at this critical number wecompute the second derivative f''(x) and find that its value at thecritical number is  , a positive number.
Thus the minimum length of cable needed is
