**III. Tangent Lines and Optimization in Business and Economics**

In business and economics, common questions of interest include how to maximize profit or revenue, how to minimize average costs, and how to maximize average productivity. For questions such as these, the description of how to obtain the desired maximum or minimum represents an optimal (or best possible) solution to the problem. And the process of finding an optimal solution is called optimization. Answering these questions often involves tangent lines. In this project, we examine how tangent lines can be used to minimize average costs.

Suppose that Wittage, Inc., manufactures paper shredders for home and office use and that its weekly total costs (in dollars) for *x* shredders are given by

C
(
x
)
=
0.03
x
2
+
12.75
x
+
6075

The average cost per unit for *x* units [denoted by
C
¯
(
x
)
] is the total cost divided by the number of units. Thus, the average cost function for Wittage, Inc., is

C
¯
(
x
)
=
C
(
x
)
x
=
0.03
x
2
+
12.75
x
+
6075
x
=
0.03
x
2
x
+
12.75
x
x
+
6075
x
C
¯
(
x
)
=
0.03
x
+
12.75
+
6075
x

Wittage, Inc., would like to know how the company can use marginal costs to gain information about minimizing average costs. To investigate this relationship, answer the following questions.

Between what pair of consecutive *x*-values in the table will
C
¯
reach its minimum?