Applied Mathematics for Business and Economics

Lecture Note Differentiation

Example 3
Use the second derivative test to find the relative maxima and minima of the function

fx()=+−− 2312 x^32 x x 7.

(Answer: relative minimum pointሺ1,െ14ሻ and relative maximum pointሺെ2,13ሻ)

Example 4
Find the point of diminishing returns for the sales function

Sx( )=−0.02x^32 + + 3 x 100

where x represents thousands of dollars spent on advertising, 0 ≤x≤ 80 and S is
sales in thousands of dollars for automobile tires.

Solution
Find the hypercritical values of x between 0 and 80, and determine whether these
points are points of inflection.

()

()

()

32

2

0.02 3 100

0.06 6

0.12 6

Sx x x

Sx x x

Sx x

=− + +

′ =− +

′′ =− +

Setting Sx′′()= 0 gives

−0.12x+=x^6 = 500

Testing will show that

( )

()

0for0 50

0 for 50 80

Sx x

Sx x

′′ ><<

′′ < <<^

The point of diminishing returns is at (50,S( (^50) ))=(50, 5100), where $50,000 is
spent on advertising, and sales in tires are $5,100,000.

8 Applications to Business and Economics

Remember that in an inventory problem total cost may include ordering cost (to
cover handling and transportation), storage cost, and purchase cost. Then we get

Total costൌstorage cos൅ordering cost൅purchase cost

Average cost per unit (AC) is the total cost divided by the number of units produced.

Hence, if Cq( )

s

denotes the total cost of producing q units of item, the average cost

per unit i ()

Cq()

AC q

q

=.

ܥܣ൏

ܥܣ ܥܣ൐

ܥܣ ܥܣൌ

We have the relationship between average cost and marginal cost which is stated as
follows:
Suppose AC and MC denote the average cost and marginal cost respectively.Then
AC is decreasing when ܥܯ
is increasing when ܥܯ
has (first-order) critical point (usually relative minimum) when ܥܯ
Students are strongley recommended to do mathematical proof for these facts.

8.1 Elasticity of Demand