(a) If C ( x ) is the cost of producing x units of a...

(a) If $C ( x )$ is the cost of producing $x$ units of a commodity, then the average cost per unit is $c ( x ) = C ( x ) / x$ . Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If $C ( x ) = 16,000 + 200 x + 4 x ^ { 3 / 2 } ,$ in dollars, find $( \mathrm { i } )$ the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum aver- age cost.

(a) If $C ( x )$ is the cost of producing $x$ units of a commodity,
then the average cost per unit is $c ( x ) = C ( x ) / x$ . Show
that if the average cost is a minimum, then the marginal
cost equals the average cost.
(b) If $C ( x ) = 16,000 + 200 x + 4 x ^ { 3 / 2 } ,$ in dollars, find $( \mathrm { i } )$ the
cost, average cost, and marginal cost at a production
level of 1000 units; (ii) the production level that will
minimize the average cost; and (iii) the minimum aver-
age cost.

Key Concepts

Cost Function

A cost function represents the total cost incurred in producing a certain number of units. It is a fundamental concept in economics and business that aggregates all expenses (fixed and variable) associated with manufacturing or providing a product or service.

Average Cost

The average cost is obtained by dividing the total cost by the number of units produced. This measure provides insight into the cost per unit and is essential for assessing pricing strategies, profitability, and efficiency in production.

Marginal Cost

Marginal cost is defined as the derivative of the cost function with respect to the number of units produced, representing the additional cost incurred by producing one more unit. It is crucial in decision-making processes such as determining optimal production levels.

Optimization Using Derivatives

Optimization in this context involves finding the production level at which the average cost is minimized. By using derivatives, one can set the derivative of the average cost function to zero to identify critical points. This process is a key application of differential calculus in economic analysis.

Relationship Between Marginal and Average Cost

At the point where the average cost is minimized, the marginal cost equals the average cost. This equality stems from the application of the quotient rule in differentiation of the average cost function and serves as an important condition in cost analysis and optimization.

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