derivative test notes

 Fermat's theorem

 If a function ƒ(x) is defined on the interval (a, b), and there exists a point
 on the interval for which x = c such that a < c < b, then if the point
 (c, ƒ(c)) is a local extremum and ƒ′(c) exists, then (c, ƒ(c)) must be a
 critical point of ƒ(x) such that ƒ′(c) = 0.

+------------------------------------------------------------------------------+
|                            FIRST DERIVATIVE TEST                             |
|                                                                              |
+------------------------------------------------------------------------------+
|                                                                              |
| Suppose that c is a point such that the first derivative is 0, f '(c) = 0    |
|                                                                              |
|     If f' changes from positive to negative at c, then c is a local maximum. |
|                                                                              |
|     If f' changes from negative to positive at c, then c is a local minimum. |
|     If f' does not change at c, no minimum/maximum exists at c.              |
|                                                                              |
+------------------------------------------------------------------------------+
|                           SECOND DERIVATIVE TEST                             |
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+------------------------------------------------------------------------------+
|                                                                              |
| Let f '' is continuous near c.                                               |
|                                                                              |
|     If f '' (c) > 0, there is a local minimum at c.                          |
|                                                                              |
|     If f '' (c) < 0, there is a local maximum at c.                          |
|                                                                              |
+------------------------------------------------------------------------------+

To find critical points, set the first derivative equal to zero and solve for
the zeros.

To find the extrema for the function f over the closed interval [a, b]:

1. Find the critical numbers of f in (a, b).

2. Evaluate f at each critical number found in Step 1 over (a, b).

3. Evaluate f at each end point of the interval [a, b].

4. The least of these values is the minimum and the greatest is the maximum.