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Second Derivative Test for Local Extrema

  1. Apr 5, 2014 #1
    Suppose ##f^{\prime\prime}## is continuous on an open interval that contains x = c

    1. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)<0##, then ##f## has local maximum at x = c.
    2. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)>0##, then ##f## has local minimum at x = c.
    3. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)=0##, then the test fails. The function ##f## may have a local maximum, a local minimum, or neither.

    I've a little doubt in point 3. I've come up with only 1 example for the possibility when

    ##f^{\prime}(c)=0##, ##f^{\prime\prime}(c)=0## and ##f## has neither local maximum or local minimum.

    Ex: ##f(x)=x^3##

    image.jpg

    Please give examples for other two possibilities when

    ##f^{\prime}(c)=0##, ##f^{\prime\prime}(c)=0## and f has local maximum

    ##f^{\prime}(c)=0## , ##f^{\prime\prime}(c)=0## and f has local minimum
     
    Last edited by a moderator: Apr 18, 2014
  2. jcsd
  3. Apr 5, 2014 #2

    UltrafastPED

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  4. Apr 5, 2014 #3

    HallsofIvy

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    For the first, [itex]f(x)= 1- x^4[/itex]. For the second, [itex]f(x)= x^4[/itex].
     
  5. Apr 5, 2014 #4
    Thanx I get it
     
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