average cost at what level of output is average cost minimum? TC= 60q-12q^2+q^3

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You need to remember that the derivative of function tells you about the monotony of a function, hence you should find derivative of total cost function such that:

TC'(q) = 60 - 12*2*q + 3q^2

You need to solve the equation TC'(q) = 0 such that:

3q^2 - 24q + 60 = 0

q^2 - 8q + 20 = 0

Notice that the function q^2 - 8q + 20 != 0 for any real value of q, hence the marginal cost given by the function TC'(q) = 3q^2 - 24q + 60 does not reach an extreme for all real values of q.

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