Finding Min/Max points for a function f(x):

 

 

First, find the critical points of f(x).
 That is, compute f’(x) and find the points c where either:

    1. ___________________.
    2. ___________________.

 

  1. For every critical point c, you can tell if it’s a local min or max by one of these tests:
    1. First Derivative Test If c is a critical point, and if f is continuous at c, then c is a:

                                                              i.      Local MAX if f ’ switches from ______________ to ___________ .

                                                            ii.      Local min if f ’ switches from ______________ to ___________ .

                                                          iii.      neither, if f ’ does not change sign at c.

    1. Second Derivative Test If f ’(c)=0 and f ’’ is continuous near c, then c is:

                                                              i.      Local MAX if f ’’(c) ________________

                                                            ii.      Local min if f ’’(c) __________________

                                                          iii.      The test FAILS (is inconclusive), if f ’’(c) =0

[c can be a local max/min, or an inflection point when f ’’ changes sign)]

  1. You can find the absolute (global) minimum and maximum (if any) by the following:
    1. Evaluate f at all the critical points.
    2. Evaluate f at all the closed endpoints of the domain, if any,
      or compute the limits at the open endpoints (or at infinity).
    3. Compare all the values you computed to determine your absolute max and min (if any).

Note: If the domain is _______________, and if the function is _____________, you’re guaranteed to have both an absolute min and an absolute max. Otherwise, you might or might not have any.