9781118041581

Appendix to Chapter 2 Calculus and Optimization Techniques 67

to negative as output increases; that is, the slope of the profit function decreases

as output increases around the maximum. In contrast, at a minimum, the slope

changes from negative to zero to positive; the slope is increasing. Because of

this difference, the secondderivative of the profit function can be used to dis-

tinguish between the two cases. The second derivative is found by taking the

derivative of d/dt. If the second derivative is negative (i.e., the slope is

decreasing), the point in question is a local maximum; if the second derivative

FIGURE 2A.2

A Second Profit

Function

The manager must be

careful to distinguish a

maximum from a

minimum.

10

8

6

4

2

0

–2

–4

–6

–8

• 10


• 12


• 14


• 16

2 4 6 8 10 12 14 16

Profit (Thousands of Dollars)

Quantity (Thousands of Units)

Quantity

(000s)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Profit

($000s)

–10.0

–15.6

–11.6

• 2.8
  6.0
  10.0
  4.4
  –15.6

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