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.