Local Vs Global Max/Min by HC11120608751

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									                               Local Vs Global Max/Min
Local Max/Min
         A local maximum/minimum is a point (p, f(p)) on the graph of that has the
largest/smallest function value around. Formally we say there is a local max/min at x = p
if f(p) is the largest/smallest function value in a neighborhood or set of points around p.
         Local maxima and minima occur at points where f   x   0 or where f   x  is
undefined. It’s important to remember that not all points where f   x   0 or is
undefined are local extrema but that all local extrema are at points where f   x   0 or is
undefined. The requirement that f   x   0 or is undefined is known as a necessary but
not sufficient condition for local extrema. It also gives us a good way to look for them.
                a.) Find f '(x)
                b.) Find all points where f '(x) = 0 or is undefined. These are known as
                the critical points (CPs) of f(x).
                c.) Test each CP to see if it is a local max/min or neither.
                d.) The first derivative test says that if f '(x) changes sign at a CP then it
                is a local max/min. If f '(x) changes from positive to negative at the CP
                then the CP is a local max, if f '(x) changes from negative to positive at the
                CP then the CP is a local min and if f '(x) doesn’t change sign at the CP
                then the CP is neither a local max nor a local min.
                e.) The second derivative test says that if f "(x) is positive at the CP then
                the CP is a local min, if f "(x) is negative at the CP then the CP is a local
                max and if f "(x) is zero at the CP the second derivative test is
                inconclusive.
                              x5 16 x3
        Example: f  x   
                              5      3
                Find the CPs:
                                                                        
                                    f   x   x 4  16 x 2  x 2 x 2  16  x 2  x  4  x  4   0
                                    when x  0 or x  4 or x  4
                 Test the CPs:
                                  Using the first derivative test, when x = 4 f   x   0 for
                                  values to the left of 4 and f   x   0 for values to the right
                                  of 4 (try evaluating f  3.9 and f   4.1 to see what
                                  happens). This tells us that there is a local minimum at
                                  x = 4.

                                  Using the second derivative test, when x = -4
                                  f   x   4x3  32x and
                                   f   4  4  4  32  4  256  128  128  0 .
                                                     3


                                  This tells us that there is a local maximum at x = -4.
                               When x = 0 the second derivative test is inconclusive
                               because f   0  0 . Using the first derivative test we see
                               that f   x   0 both to the right of 0 and to the left of 0.
                               (try evaluating f   .1 and f  .1 to see what happens)
                               This tells us that x = 0 is neither a local max nor a local
                               min.

Global Max/Min
        A global max/min is a point (p, f(p)) on the graph of f(x) that has the
largest/smallest function value on the domain of f(x). Formally we say there is a global
max at x = p if f  p   f  x  for all x in the domain and a global minimum at x = p if
 f  p   f  x  for all x in the domain.
          Any function that is continuous on a closed, bounded interval [a, b] will have both
a global max and a global min on that interval. To find them
                   a.) Find all the CPs in [a, b]
                   b.) Find f(a) , f(b) and f(CP) for all CPs in [a, b]
                   c.) The largest of these function values will be the global max and the
                   smallest will be the global min. It’s important to remember that the global
                   max/min might be at one of the endpoints or it might be at one of the CPs
                   but it will exist.

                                                                         x5 16 x3
       Example: Find the global max and min of the function f  x               on the
                                                                         5      3
               interval 7,5 . We already know there is a local max at x = -4 and a
               local min at x = 4. We want to find the function values at thest CPs and
               also at the endpoints:
                                      f  7   1532
                                        f  4   136.53
                                        f  4   136.53
                                        f  5   41.67
               We can see that on the interval 7,5 the global max occurs at x = -4 and
               the global min at x = -7.

								
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