   Chapter 3.7, Problem 59E

Chapter
Section
Textbook Problem

# (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 (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 average cost.

To determine

a)

To show:

If the average cost is a minimum then the marginal cost equals the average cost

Explanation

1) Concept:

At a minimum c'(x)=0

2) Given:

Average cost c(x)=C(x)x

3) Calculation:

Given that

c(x)=C(x)x

Differentiate c(x) using quotient rule

c'x=Cxx'

ddxCxx=x·ddxCx-Cx·ddxxx2

At a minimum c'(x)=0

x·ddxCx-Cx·ddxxx2=0

Simplify

x·C'x-C<

To determine

b)

To find:

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

iii) The minimum average cost

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Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach 