AP CALCULUS AB

Chapter 4:Applications of Derivatives

Section 4.1:Extreme Values of Functions

What you’ll learn about Absolute (Global) Extreme Values Local (Relative) Extreme Values Finding Extreme Values

…and whyFinding maximum and minimum values of afunction, called optimization, is an importantissue in real-world problems.

Absolute Extreme Values Let be a function with domain . Then ( ) is the

on if and only if ( ) ( ) for all in .

on if and only if ( ) ( ) for all in .

f D f c

D f x f c x D

D f x f c x D

(a) absolute maximum value

(b) absolute minimum value

Absolute maximum and minimum values are also called absolute extrema. The term “absolute” is often omitted.

The Extreme Value Theorem If is continuous on a closed interval [ , ], then has both a maximum

value and a minimum value on the interval.

f a b f

The Extreme Value TheoremIf f is continuous on a closed interval [a,b], then f has

both a maximum value and a minimum value on the interval.

2 conditions for f: continuous & closed interval

If either condition does not exist the E.V.T. does not apply.

Classifying Extreme Values

Local Extreme Values Let be an interior point of the domain of the function . Then ( ) is a

at if and only if ( ) ( ) for

all in some open interval containing .

c f f c

c f x f c

x c

(a) local (or relative) maximum value

(b) local ( at if and only if ( ) ( ) for

all in some open interval containing .

A function has a local maximum or local minimum at an endpoint if

the appropriate inequality h

c f x f c

x c

f c

or relative) minimum value

olds for all in some half-open domain

interval containing .

x

c

Local Extreme Values If a function has a local maximum value or a local minimum value at

an interior point of its domain, and if ' exists at , then '( ) 0.

f

c f c f c

Extreme Values can be Absolute or Local Local (relative) Extreme ValuesLet c be an interior point of the domain of the function f. Then f(c) is a a) Local maximum value at c if and only if f(x) ≤ f(c) for all x in some open interval containing c.b) Local minimum value at c if and only if f(x) ≥ f(c) for all x in some open interval containing c.

A function f may have a local max or local min at an endpoint c if the appropriate inequality holds.

An extreme value may be local and global.

Critical Points A point in the interior of the domain of a function at which ' 0 or ' does

not exist is a of .

f f f

f

critical point

Section 4.1 – Extreme Values of Functions Not all critical numbers (points) may be

actual relative minimums or maximums.

minimum wheref’(c) does not exist

f’(c)=0, butno max or min

Example Finding Absolute Extrema

2 / 3Find the absolute maximum and minimum values of ( ) 3 on the

interval [-1, 2].

f x x

3

2 / 3

2Find the critical point values: '( ) has no zeros but is undefined at 0.

Critical point value: (0) 0

Endpoint values: ( 1) 3;

(2) 3 2 4.762

The absolute maximum value

f x xx

f

f

f

is 4.762 and occurs at 2.

The absolute minimum value is 0 and occurs at 0.

x

x

Example Finding Extreme Values 2

1Find the extreme values of ( ) .

9f x

x

2

-1/ 22

2

3 / 22

3 / 22

is defined for 9 - 0, so its domain is the interval ( 3,3). Since the domain

has no endpoints, all the extreme values must occur at critical points.

1( ) 9

9 -1

'( ) 9 22 9

T

f x

f x xx

xf x x x

x

he only critical point is at 0. The only candidate for an extreme value is

(0) 1/ 3. To determine whether 1/3 is an extreme value of , examine ( ).

As moves away from 0 on either side, the denom

x

f f f x

x

inator gets smaller, the

values of increase and the graph rises. There is an absolute minimum at

0. The function has no maxima, either local or absolute.

f

x

Section 4.1 – Extreme Values of Functions Theorem: Local Extreme Values

If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if exists at c, thenIn other words, if f has a relative maximum or relative minimum at x = c, then c is a critical number of f.

f .0 cf

Section 4.1 – Extreme Values of Functions Finding Extrema on a Closed Interval1. Find the critical numbers of f in 2. Evaluate f at each critical number in 3. Evaluate f at each endpoint of4. The least of these values is the absolute

minimum.5. The greatest of these values is the

absolute maximum.

ba, ba,

ba, ba,

(Absolute)

GraphicallyLook at the graph.

The highest point on the graph is the global / absolute maximum.

The lowest point on the graph is the global / absolute minimum.

Don’t forget to consider the endpoints when looking at a closed interval!

Find the maximum and minimum points of f(x) = cos x on [-π, π]

Example 3 Finding Absolute ExtremaFind the absolute maximum and minimum values of

f(x) = x 2/3 on the interval [-2,3].

Find x values where f ’ = 0 (make sure they are in the domain) Check endpoints of domain Find the y-values of each critical point to determine the maximum and minimum

points.

Practice: Find absolute extrema of g(x)=e-x on [-1,1].

Finding Extreme ValuesFind the extreme values of

Derivative = 0EndpointsFind y-values to evaluate

Find the extreme values of

24

1)(

xxf

1,2

1,25)(

2

xx

xxxf

Make Sure to Answer the Question being Asked! What is the maximum value? The output value is crucial.

Where does the maximum occur? The input value is crucial.

Where on the curve is the maximum? The POINT (ordered pair) is crucial.

Summary Absolute extrema is on a closed interval

and can be at the critical numbers (derivative = 0 or undefined) or at the endpoints of the interval.

Relative/local extrema are the maximum/minimum values that the function takes on over smaller open intervals. There are no endpoints to test like we do for absolute extrema.

You try: What are the extreme values of

on , ?2 2

1) 3cos3

f x

f x x

You try: Find the absolute maximum and minimum

values of on the interval 2, 2 .xf x xe

Section 4.1 – Extreme Values of Functions Finding the Relative (local) maximum or

minimum of a function (using a graphing calculator)

1. Enter the equation in y=.2. Graph in the appropriate window.3. Use (2nd) CALC, maximum or minimum (On the

TI-89, use F5 (Math), max or min).4. Arrow to just left of the critical point, ENTER.5. Arrow to just right of the critical point, ENTER.6. Guess? – Just press ENTER

Section 4.1 – Extreme Values of Functions To find approximations of the roots of a

polynomial using a graphing calculator:1. Enter the equation in y=.2. Graph in the appropriate window.3. Use (2nd) CALC, ZERO (on the TI-89, use F

% (Math) ZERO).4. Arrow to just left of the zero, ENTER.5. Arrow to just right of the zero, ENTER.6. GUESS? – press ENTER.