# Math

Given that the total cost of a firm is C(q)= 60q- 12q^2+ q^3

• Derive an expression for marginal cost
• Derive an expression for average cost
• At what level of output is average cost minimum?
• 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 C(q)= 60q - 12q^2 + q^3

The marginal cost is C'(q) = 60 - 24q + 3q^2

The average cost is C(q)/q = (60q - 12q^2 + q^3)/q = 60 - 12q + q^2

The derivative of 60 - 12q + q^2 is...

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The total cost of a firm is C(q)= 60q - 12q^2 + q^3

The marginal cost is C'(q) = 60 - 24q + 3q^2

The average cost is C(q)/q = (60q - 12q^2 + q^3)/q = 60 - 12q + q^2

The derivative of 60 - 12q + q^2 is -12 + 2q

The average cost is minimum when -12 + 2q = 0

=> q = 6

At q = 6, the average cost is 60 - 12*6 + 6^2 = 24 and the marginal cost is equal to 60 - 24*6 + 3*6^2 = 24.

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