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TWEET # derivative test notes a guest Nov 17th, 2019 68 Never
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1.  Fermat's theorem
2.
3.  If a function ƒ(x) is defined on the interval (a, b), and there exists a point
4.  on the interval for which x = c such that a < c < b, then if the point
5.  (c, ƒ(c)) is a local extremum and ƒ′(c) exists, then (c, ƒ(c)) must be a
6.  critical point of ƒ(x) such that ƒ′(c) = 0.
7.
8. +------------------------------------------------------------------------------+
9. |                            FIRST DERIVATIVE TEST                             |
10. |                                                                              |
11. +------------------------------------------------------------------------------+
12. |                                                                              |
13. | Suppose that c is a point such that the first derivative is 0, f '(c) = 0    |
14. |                                                                              |
15. |     If f' changes from positive to negative at c, then c is a local maximum. |
16. |                                                                              |
17. |     If f' changes from negative to positive at c, then c is a local minimum. |
18. |     If f' does not change at c, no minimum/maximum exists at c.              |
19. |                                                                              |
20. +------------------------------------------------------------------------------+
21. |                           SECOND DERIVATIVE TEST                             |
22. |                                                                              |
23. +------------------------------------------------------------------------------+
24. |                                                                              |
25. | Let f '' is continuous near c.                                               |
26. |                                                                              |
27. |     If f '' (c) > 0, there is a local minimum at c.                          |
28. |                                                                              |
29. |     If f '' (c) < 0, there is a local maximum at c.                          |
30. |                                                                              |
31. +------------------------------------------------------------------------------+
32.
33. To find critical points, set the first derivative equal to zero and solve for
34. the zeros.
35.
36. To find the extrema for the function f over the closed interval [a, b]:
37.
38. 1. Find the critical numbers of f in (a, b).
39.
40. 2. Evaluate f at each critical number found in Step 1 over (a, b).
41.
42. 3. Evaluate f at each end point of the interval [a, b].
43.
44. 4. The least of these values is the minimum and the greatest is the maximum.
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