Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby to

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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 ( 11 , 0 ) in the x y -plane, Springfield is at ( 0 , 3) , and Shelbyville is at ( 0 , − 3 ) . The cable runs fromCenterville to some point ( x , 0 ) on the x -axis where it splitsinto two branches going to Springfield and Shelbyville. Find thelocation ( x , 0 ) that will minimize the amount of cable betweenthe 3 towns and compute the amount of cable needed. Justify youranswer.
To solve this problem we need to minimize the following functionof x :
f ( x ) = ?
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 itsvalue at the critical number is: ?
Thus the minimum length of cable needed is: ?