The total cost function of a firm that produces its product on two assembly lines is given as  TC (x,y) = 3x2+6y2-xy The problem facing the firm is to detrmine the least-cost combination of output on assembly lines x and y subject to the side condition that total output equal 20 units.  Find the out put of x and y subject to a total of 20 units that minimize total cost by using both substitution and Lagrangian multiplier methods

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