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Math Help - Determining the type of critical point when the second derivative test fails

  1. #1
    Junior Member
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    Aug 2009
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    Determining the type of critical point when the second derivative test fails

    For example with f(x,y) = x^2y + xy^2
    Well I know there is a critical point at (0,0). So I calculated the second derivatives but they are all 0 here so that doesn't help.
    I also tried using the Taylor expansion to show that f(x,y)>f(0,0) or not but that didn't get me anywhere.
    Then I tried considering the type of critical point on x=0,y=0,y=-x etc. but again I didn't get anywhere.
    Any ideas?
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  2. #2
    Senior Member
    Joined
    Mar 2010
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    f(x,y) = x^2y + xy^2.

    If y=x
    f(x,y) = 2x^3
    if x<0 f(x,y)<0
    if x>0 f(x,y)>0.
    (0,0) is not min point.
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