Sec4.1: MAXIMUM AND MINIMUM VALUES

Sec4.1: MAXIMUM AND MINIMUM VALUES

absolute maximum

global maximum

local maximumrelative maximum

How many local maximum ??

Sec4.1: MAXIMUM AND MINIMUM VALUES

absolute minimumglobal minimum

local minimumrelative minimum

How many local minimum ??

The number f(c) is called the maximum value of f on Df(c)

c d

The number f(d) is called the maximum value of f on D

f(d)

The maximum and minimum values of are called the extreme values of f.

c2 c1

Example1:

Example2:

Example3:

We have seen that some functions have extreme values, whereas others do not.

1 f(x) is continuous on

2 Closed interval [a, b]

attains an absolute maximum f(c) and minimum f(d) value

],[, badc

THE EXTREME VALUE THEOREM

1 f(x) is continuous on

2 Closed interval [a, b]

attains an absolute maximum f(c) and minimum f(d) value

],[, badc

THE EXTREME VALUE THEOREM

Max?? Min?? What cond??Max?? Min?? What cond??

11

Remark:The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values.

1 f(x) is continuous on

2 Closed interval [a, b]

attains an absolute maximum f(c) and minimum f(d) value

],[, badc

THE EXTREME VALUE THEOREM

Fermat’s Theorem is named after Pierre Fermat (1601–1665), a French lawyer who took up mathematics as a hobby. Despite his amateur status, Fermat was one of the two inventors of analytic geometry (Descartes was the other). His methods for finding tangents to curves and maximum and minimum values (before the inventionof limits and derivatives) made him a forerunner of Newton in the creation of differential calculus.

Sec 3.11 HYPERBOLIC FUNCTIONSThe following examples caution us against reading too much into Fermat’s Theorem. We can’t expect to locate extreme values simply by setting f’(x) = 0 and solving for x.

Exampe5: 3)( xxf Exampe6: xxf )(

WARNING Examples 5 and 6 show that we must be careful when using Fermat’s Theorem. Example 5 demonstrates that even when f’(c)=0 there need not be a maximum or minimum at . (In other words, the converse of Fermat’s Theorem is false in general.) Furthermore, there may be an extreme value even when f’(c)=0 does not exist (as in Example 6).

F092

F081

F083

F091

In terms of critical numbers, Fermat’s Theorem can be rephrased as follows (compare Definition 6 with Theorem 4):

F092

F091

F081

F081

F092

F081

F083

F083