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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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