OPTIMIZATION IN BUSINESS/ECONOMICS

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Presentation transcript:

OPTIMIZATION IN BUSINESS/ECONOMICS using derivatives

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

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

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?

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

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]

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

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

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