OPTIMIZATION IN BUSINESSECONOMICS using derivatives EXTREME VALUES AbsoluteGlobal

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