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Second derivative test

  1. Jun 6, 2015 #1
    According to this link: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx
    The second derivative test can only be applied if ##f''## is continuous in a region around ##c##.
    But according to this link: http://calculus.subwiki.org/wiki/Second_derivative_test#Requirement_of_twice_differentiability
    ##f''## need not be defined in a region around ##c##.
    I'm confused as to what is required for the second derivative test.
    ##f''## is allowed to have a discontinuity at ##c##, but not around ##c##? If ##f''## is continuous around ##c##, then ##\lim_{x→a} f''(x)## must equal ##f''(a)## where ##a## is in some region around ##c##, but according to the second link, ##f''(a)## doesn't have to be defined.
     
  2. jcsd
  3. Jun 6, 2015 #2

    wabbit

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    There is no need for ## f''## to be defined at points other than c. If ## f'(c)=0 ## and ## f''(c)>0 ##, using the definition ## f''(c)=\lim_{h\rightarrow 0}\frac{f'(c+h)-f'(c)}{h} ## this implies that within some neighborhood of ## c,f' ## is positive to the right of of ## c ## and negative to its left, so f must have a minimum at ## c ## .
     
  4. Jun 6, 2015 #3
    http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx#Extras_DerAppPf_SDT

    The author assumes that ##f''## is continuous around ##c## to prove the second derivative test.
     
  5. Jun 6, 2015 #4
    Only that ##f''## is defined at ##c##. I don't see where in the first link it suggests a requirement for continuity: if it does it is wrong.
     
  6. Jun 6, 2015 #5

    wabbit

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    I just gave you a proof that does not require that assumption. Did you read it ? If you don't understand it I can help you with fleshing out the details.

    Whatever the author assumes is irrelevant to the question you asked about which assumption is necessary.
     
    Last edited: Jun 6, 2015
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