Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. Centerville is located at (7,0) in the xy-plane, Springfield is at (0,4), and Shelbyville is at (0,-4). To save on the cost of cable, Greedy Cablevision wants to arrange the cable in a Y-shaped configuation, running cable from Centerville to some point (x,0) on the x-axis where it then splits into two branches, one going to Springfield and one to Shelbyville.

Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed.

(a) To solve this problem we need to minimize the following function of x:

f(x)=
(b) We find that f(x) has a critical point at

x=
.

(c) To verify that f(x) has a minimum at this critical point we evaluate the second derivative f′′(x) at this point.

f′′(critical point) is
, a positive number.

(d) Thus the minimum length of cable needed is