3.1 extrema on an interval. after this lesson, you should be able to: understand the definition of...

3.1 Extrema On An Interval

After this lesson, you should be able to:

Understand the definition of extrema of a function on an intervalUnderstand the definition of relative extrema of a function on an open intervalFind extrema on a closed interval

Definition

When the just word minimum or maximum is used, we assume it’s an absolute min or absolute max.

ExtremaMinimum and maximum values on an interval are called extremes, or extrema on an interval.• The minimum value of the function on an interval is considered the absolute minimum on the interval.• The maximum value of the function on an interval is considered the absolute maximum on the interval.

OPEN intervals – Do the following have extrema?

On an open interval, the max. or the min. may or may not exist even if the function is continuous on this interval.

CLOSED intervals – Do the following have extrema?

On a closed interval, both max. and min. exist if the function is continuous on this interval.

The Extreme Value Theorem (EVT)Theorem 3.1: If f is continuous on a closed interval [a, b], then f has both a minimum and a maximum on the interval.In other words, if f is continuous on a closed interval, f must have a min and a max value.Max-Min

f is continuous on [a, b]

a b

Example

Example 1 Let f (x) = x2 – 5x – 6 on the closed interval [–1, 6], find the extreme values.

Example

Example 2 Let f (x) = x3 + 2x2 – x – 2 on the closed interval [–3, 1], find the extreme values.

Example

Example 3 Let f (x) = x3 + 2x2 – x – 2 on the closed interval [–3, 2], find the extreme values.

The (absolute)max and (absolute)min of f on [a, b] occur either at an endpoint of [a, b] or at a point in (a, b).

Relative Extrema and Critical Numbers

(AP may use Local Extrema)

1. If there is an open interval containing c on which f (c) is a maximum, then f (c) is a local maximum of f.

2. If there is an open interval containing c on which f (c) is a minimum, then f (c) is a local minimum of f.

When you look at the entire graph (domain), there may be no absolute extrema, but there could be many relative extrema.

What is the slope at each extreme value????

Definition of a Critical Number and Figure 3.4

Critical Numbersc is a critical number for f iff:

1. f (c) is defined (c is in the domain of f )2. f ’(c) = 0 or f ’(c) = does not exist

Theorem 3.2 If f has a relative max. or relative min, at x = c, then c must be a critical number for f.

The (absolute)max and (absolute)min of f on [a, b] occur either at an endpoint of [a, b] or at a critical number in (a, b).

So…. Relative extrema can only occur at critical values, but not all critical values are extrema. Explain this statement.Explain this statement.

Make sure f is continuous on [a, b].

1. Find the critical numbers of f(x) in (a, b). This is where the derivative = 0 or is undefined.

2. Evaluate f(x) at each critical numbers in (a, b).

3. Evaluate f(x) at each endpoint in [a, b].

4. The least of these values (outputs) is the minimum. The greatest is the maximum.

**Make sure you give the y-value this is the extreme value!**

Guidelines

Critical Numbers

1. Make sure f is continuous on [a, b].2. Find all critical numbers c1, c2, c3…cn of

f which are in (a, b) where f’(x) = 0 or f’(x) is undefined.

3. Evaluate f(a), f(b), f(c1), f(c2), …f(cn).4. The largest and smallest values in part

2 are the max and min of f on [a, b].

To find the max and min of f on [a, b]:

Example

60426 2 xx

Example 4 Find all critical numbers

460212)( 23 xxxxf

Domain:

)(' xf )5)(2(6 xx

(–, +)

Critical number:

x = 2 and x = 5

Example

2

2

2 ( 1)

( 1)

x x x

x

2

2 2

2 ( 2)

( 1) ( 1)

x x x x

x x

Example 5 Find all critical numbers. 1

)(2

x

xxf

Domain:

)(' xf

x ≠ 1, xR

Critical number:

x = 1, x = 0, and x = 2

existnot does )1(' ,0)2(')0(' fff

ExampleExample 6 Find all critical numbers.

3

2

)4()( xxf

Domain:

)(' xf

(–, +)

3 4 3

2

x

Critical number:

x = –4

existnot does )4(' f

f’(–4 )

Example

x Left Endpoint

Critical Number

Critical Number

Right Endpoint

f (x)

f (–3)= 20 f (–2)= 30 f (4)=–78 f (5)=–68

)4)(2(32463 2 xxxx

Example 7 Find the max and min of f on the interval [–3, 5]. 2243)( 23 xxxxfDomain:

)(' xf

(–, +)

Critical number:

x = –2 and x = 4

minimummaximum

Graph is not in scale

Practice Of

x Left Endpoint

Critical Number

Critical Number

Right Endpoint

f (x)

f (–1)= 7 f (0)= 0 f (1)=–1 f (2)=16

)1(121212 223 xxxx

Example 7 Find the extrema of f on the interval [–1, 2]. 34 43)( xxxf Domain:

)(' xf

(–, +)

Critical number:

x = 0 and x = 1

minimum maximum

ExampleExample 8 Find the extrema of f on the interval [–1, 3].

3

1

3

1

3

1

12

22)('

x

x

xxf

3

2

32)( xxxf

Critical number:

x = 0 and x = 1

x Left Endpoint

Critical Number

Critical Number

Right Endpoint

f (x)

f (–1)= –5 f (0)= 0 f (1)=–1 f (3)=

minimum maximum

24.09 36 3

f ’(0) does not exist

Practice OfExample 8 Find the extrema of f on the interval [0, 2].

xxxf 2cossin2)( xxxf 2sin2cos2)('

xxx cossin4cos2 )sin21(cos2 xx

Critical number:

x = /2, x = 3/2, x = 7/6, x = 11/6 x Left

Endpoint

Critical Number

Critical Number

Critical Number

Critical Number

Right Endpoint

f (x)

f (0)=–1

f (/2)=

3

f (7/6) =–3/2

f (3/2) =–1

f (11/6) =–3/2

f (2) =–1

minmax min

Summary

Open vs. Closed Intervals1. An open interval MAY have extrema2. A closed interval on a continuous curve will ALWAYS have a minimum and a maximum value. 3. The min & max may be the same value How?

Homework

Section 3.1 page 169 #1,2,13-16,19-24,36,60