Transcript
![Page 1: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/1.jpg)
Sec 3.4: Concavity and the Second Derivative Test
•Determine intervals on which a function is concave upward or concave downward.•Find any points of inflection of the graph of a function.•Apply the Second Derivative Test to find relative extrema of a function.
![Page 2: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/2.jpg)
Definition of Concavity
![Page 3: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/3.jpg)
![Page 4: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/4.jpg)
![Page 5: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/5.jpg)
![Page 6: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/6.jpg)
Ex01: Determining Concavity
3
6)(
2 x
xf
![Page 7: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/7.jpg)
Ex02: Determining Concavity
4
1)(
2
2
x
xxf
![Page 8: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/8.jpg)
Points of Inflection
![Page 9: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/9.jpg)
Ex03: Finding Points of Inflection
34 4)( xxxf
Determine the points of inflection and discuss the concavity of the graph of
![Page 10: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/10.jpg)
![Page 11: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/11.jpg)
Ex04: Using the Second Derivative Test
35 53)( xxxf
Find the relative extrema for
![Page 12: Sec 3.4: Concavity and the Second Derivative Test Determine intervals on which a function is concave upward or concave downward. Find any points of inflection](https://reader034.vdocuments.mx/reader034/viewer/2022050713/56649d765503460f94a572de/html5/thumbnails/12.jpg)
3.4 Problem Set A
• 1, 4, 7• 10, 13, 16, 19• 25, 27• 30, 33, 36, 39