A monopolist produces the same product in two factories, A and B. Total costs are as follows:

, 2A A A Q Q C + =

, 2B B B Q 5.1Q 6C +=

where Q represents the quantity of the product. The total quantity is . The inverse demand function for the product is B A T Q Q Q + =

. T Q 256 P =

What level of output should the monopolist produce in each factory in order to maximize profits? Prove that the solution you have found is actually a local maximum.