1. OPTIMIZATION IN BUSINESS/ECONOMICS
using derivatives

2. EXTREME VALUES Absolute/Global maximum of f: 4
Absolute/Global minimum of f: -4
Local maximum of f: 1
Local minimum of f: -1
A is a maximum turning point.
B is a minimum turning point.
A, B, C,and D are extreme points. A and B are local extremes. C and D are global extremes.
4, -4, 1, and -1 are extreme values

3. DO EXTREME VALUES ALWAYS EXIST?
No.
Some functions do not have extreme values.

4. EXISTENCE AND LOCATIONS OF EXTREME VALUES
If function f is continuous on a closed interval I then f has both extreme values.
If f(c) is an extreme value on I then c must be a critical point.
What is critical point?

5. TYPES OF CRITICAL POINTS (1)
Stationary points
Endpoints of closed intervals
Singular points

6. TYPES OF CRITICAL POINTS (2)
A, B are stationary points. At A, f’(0) = 0. At B, f’(2) = 0. In general: if c is a stationary point then f’(c) = 0.
C, D are endpoints of the closed interval [-1½,3½]

7. TYPES OF CRITICAL POINTS (3)
K is a singular point.
At K the derivative doesn’t exist

8. TYPES OF STATIONARY POINTS
Maximum turning points
Minimum turning points
Inflection points

9. TYPES OF STATIONARY POINTS
A is a maximum turning point. At A: f’ = 0 and f” < 0
B is an inflection point. At B: f’ = 0 and f” = 0
C is a minimum turning point. At C: f’ = 0 and f” > 0