Final answer:
To find the output level at which the average cost curve reaches its minimum, set the marginal cost functions equal to the average cost functions and solve for output. This will give us the output level at which the average cost curve of each plant reaches its minimum.
Step-by-step explanation:
The average cost curve of a production plant reaches its minimum where the marginal cost curve intersects the average cost curve. To find the output level at which this occurs, we need to find the marginal cost functions for each production plant. The marginal cost function is the derivative of the cost function with respect to output, so for the first plant, the marginal cost function is MC₁=10-8Q₁+3Q²₁ and for the second plant, the marginal cost function is MC₂=10-4Q₂+3Q²₂. To find the output level at which the average cost curve reaches its minimum, we can set the marginal cost functions equal to the average cost functions and solve for output.
a) For the first plant, setting MC₁ equal to the average cost function C₁/Q₁ gives us 10-8Q₁+3Q²₁=10Q₁ - 4Q²₁ +Q³₁. Solving this equation will give us the output level at which the average cost curve of the first plant reaches its minimum.
b) For the second plant, setting MC₂ equal to the average cost function C₂/Q₂ gives us 10-4Q₂+3Q²₂=10Q₂ - 2Q²₂ +Q³₂. Solving this equation will give us the output level at which the average cost curve of the second plant reaches its minimum.