Given  that total cost of a firm is  C(q)=60q-12q^2  + q^3.    Verify that at the same level of output at which average cost is minimum, average cost is equal to marginal cost.  

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The total cost of a firm is given in terms of the output q is C(q) = 60q - 12q^2 + q^3.

The average for the output is C(q)/q = (60q - 12q^2 + q^3)/q = 60 - 12q + q^2.

To determine the minimum value of average cost A = 60 - 12q + q^2, A' = 0 has to be solved.

A' = -12 + 2q

-12 + 2q = 0

=> q = 6

The minimum average cost is 60 - 12*6 + 36 = 24

The marginal cost is given by C'(q) and is equal to 60 - 24q + 3q^2. For q = 6, C'(q) = 60 - 24*6 + 3*36 = 24.

This shows that the the level of output where average cost is minimum the average cost is equal to the marginal cost.

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