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Applications of the First and Second Derivative

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Applications of the First and Second Derivative Powered By Docstoc
					                APPLICATIONS OF THE FIRST AND SECOND DERIVATIVE

                  =0                             >0                                   <0
f (x)     Critical Point(s)             f (x) increasing                    f (x) decreasing


f (x)   Inflection Point(s)         f(x) Concave up                    f(x) Concave down
                                    Relative Minimum(s)                 Relative Maximum(s)


Notes:

   For polynomial functions there are at most n – 1 relative minima + relative maxima
   For polynomial functions there are at most n – 2 inflection points
   Increasing/decreasing Intervals are determined by the critical points and using test points for f (x)
   Concavity Intervals are determined by the inflection points and using test points for f (x)




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                 APPLICATIONS OF THE FIRST AND SECOND DERIVATIVE
Example I: f ( x)  2 x - x  3 has at most 2 relative maxima + relative minima and at most 1 inflection point
                        3     2



             f ( x)  6 x 2 - 2x      f ( x)  12x - 2
               1. Critical points at x=0 and x = 1/3
               2. Inflection point at x = 1/6
               3. f(x) increasing on - ,0  & (1 / 3, )
                  f(x) decreasing on 0,1/3 
               4. f(x) Concave down on - ,1/6 
                          Relative maximum at x = 0
                  f(x) Concave up on 1/6,  
                          Relative minimum x = 1/3



Example II: f ( x)  x has at most 2 relative maxima + relative minima and at most 1 inflection point
                       3



             f ( x)  3x 2       f ( x)  6 x
               1. Possible critical point at x=0
               2. Inflection point at x = 0
               3. f (x) is positive everywhere therefore
                  f(x) is Increasing on - ,  
               4. Since f(x) is always increasing there is no critical
                  point, no concavity, and no relative max or min.

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                 APPLICATIONS OF THE FIRST AND SECOND DERIVATIVE


Example III: f ( x)  2 x  3 x  3 has at most 1 relative maxima + relative minima and no inflection point
                          2



             f ( x)  4 x  3   f ( x)  4
               1. Possible critical point at x=3/4
               2. -4≠0 therefore no Inflection point
               3. f(x) is Increasing on - ,3/4 
                  f(x) is decreasing on 3/4,  
               4. Since f (x) is always negative f(x) is concave down
                  everywhere - ,   and has a relative max at x=3/4




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posted:10/31/2011
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