a. Finding the output levels minimizing the average cost

First, we find the average cost for each plant by dividing the total cost by the quantity. Hence,

AC1=10−4q+q2AC1=10−4q+q2 and AC2=10−2q+q2AC2=10−2q+q2

We can now find the quantities minimizing the average costs by taking the derivative of each average cost function with respect to quantity and equating it to zero. Then, for Plant 1, we obtain q1=−−42=2q1=−−42=2 and for Plant 2, it is q2=−−22=1q2=−−22=1

b. Finding the optimal output level at each plant

Let xx be the number of units produced at Plant 1. Then, Plant 2 will produce 4−x4−x units. The company needs to minimize the total cost:

C1+C2=10x−4x2+x3+10(4−x)−2(4−x)2+(4−x)3C1+C2=10x−4x2+x3+10(4−x)−2(4−x)2+(4−x)3

Taking the derivative with respect to xx and setting it equal to zero, we obtain

10−8x+3x2−40+4(4−x)−3(4−x)2=010−8x+3x2−40+4(4−x)−3(4−x)2=0

Solving the above for xx, we find x=2.67x=2.67. Hence, 2.67 units must be produced at Plant 1 and 1.33 units - at Plant B.