3.1 extrema on an interval
DESCRIPTION
3.1 Extrema On An Interval. After this lesson, you should be able to:. Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval. Definition. Extrema. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/1.jpg)
3.1 Extrema On An Interval
![Page 2: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/2.jpg)
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
![Page 3: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/3.jpg)
Definition
![Page 4: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/4.jpg)
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.
![Page 5: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/5.jpg)
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.
![Page 6: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/6.jpg)
CLOSED intervals – Do the following have extrema?
On a closed interval, both max. and min. exist if the function is continuous on this interval.
![Page 7: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/7.jpg)
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
![Page 8: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/8.jpg)
Example
Example 1 Let f (x) = x2 – 5x – 6 on the closed interval [–1, 6], find the extreme values.
![Page 9: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/9.jpg)
Example
Example 2 Let f (x) = x3 + 2x2 – x – 2 on the closed interval [–3, 1], find the extreme values.
![Page 10: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/10.jpg)
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).
![Page 11: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/11.jpg)
Relative Extrema and Critical Numbers
(AP may use Local Extrema)
![Page 12: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/12.jpg)
![Page 13: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/13.jpg)
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????
![Page 14: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/14.jpg)
Definition of a Critical Number and Figure 3.4
![Page 15: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/15.jpg)
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.
![Page 16: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/16.jpg)
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
![Page 17: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/17.jpg)
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]:
![Page 18: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/18.jpg)
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
![Page 19: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/19.jpg)
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
![Page 20: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/20.jpg)
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 )
![Page 21: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/21.jpg)
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
![Page 22: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/22.jpg)
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
![Page 23: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/23.jpg)
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
![Page 24: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/24.jpg)
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
![Page 25: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/25.jpg)
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?
![Page 26: 3.1 Extrema On An Interval](https://reader036.vdocuments.net/reader036/viewer/2022062321/56813bbc550346895da4ebde/html5/thumbnails/26.jpg)
Homework
Section 3.1 page 169 #1,2,13-16,19-24,36,60