Transcript
![Page 1: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/1.jpg)
1
Section 4.1
Maxima and Minima
![Page 2: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/2.jpg)
2
![Page 3: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/3.jpg)
3
![Page 4: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/4.jpg)
4
a. Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x = c.
Absolute maximum at x = c
![Page 5: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/5.jpg)
5
![Page 6: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/6.jpg)
6
Absolute maximum: q and sLocal maximum: q and s
Absolute minimum: pLocal minimum: p and r
If f has a local maximum or minimum at cand f’(c) exists, then f’(c) = 0.
![Page 7: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/7.jpg)
7
Note the converse of Theorem is not necessarily true.
![Page 8: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/8.jpg)
8
X(2ln x + 1) = 0; x = 0 not in domain
or 2ln x + 1 = 0 ln x = -1/2 x = e-1/2 ≈ 0.61
![Page 9: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/9.jpg)
9
![Page 10: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/10.jpg)
10
f(-2) = 32 absolute maximum
f(3/2) = -27/16 absolute minimum
a. 22 0 2 3 0
20
3
x x
x x critical values
![Page 11: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/11.jpg)
11
b.
g(-1) = 3 absolute maximum and g(0) = g(2) = 0 absolute minimum
![Page 12: Section 4.1 Maxima and Minima 1. 2 3 4 a.Satisfies the conditions of the Extreme Value Theorem. Absolute maximum at x = a and absolute minimum at x](https://reader036.vdocuments.site/reader036/viewer/2022070323/56649dd05503460f94ac6398/html5/thumbnails/12.jpg)
12
SOLUTION