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Implicit Differentiation for a 2nd Derivative

  1. Feb 24, 2013 #1
    Hello! As of right now (10:13 PM), I've tried 9 combinations of points to solve this problem. It's a WebWork-based problem that's due in about an hour in a half. Any help would be very, very appreciated.

    1. The problem statement, all variables and given/known data

    I was given this equation: ##ln(2y) = 2xy## and was asked to find the first and second derivative from implicit differentiation.

    The first is ##2y/(y^-1-2x)##

    The second is ##4 y^3 (-3+4xy)/(-1+2xy)^3##

    These are both correct according to WebWork.

    Now, for the final part of the problem, it wants me to find ##d^2y/dx^2 = 0## at ##(x,y) = (?, ?)##

    3. The attempt at a solution

    - 3/(4x) was what I thought the right solution because it makes the numerator equal to zero, but WebWork wants only answers without variables
    - Setting ##4 y^3 (-3+4xy)/(-1+2xy)^3## equal to zero and attempting to separate variables. Unsuccessful attempt.
    - Messed around with WolframAlpha... also unsuccessful.
    - Talked to friends in higher levels of math. No fruits either.

    Could someone smarter than I provide some advice?
     
  2. jcsd
  3. Feb 24, 2013 #2

    SammyS

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    I'll give it a whirl anyway.


    You're correct in that y'' = 0 when y = -3/(4x) .

    Parametrize that. Let x = a, then y'' = 0, at (x,y) = (a, -3/(4a)) .


    (There is another set of solutions too.)
     
  4. Feb 24, 2013 #3
    Hmmm... no dice.

    I tried (a, -3/(4a)), (1, -3/(4)), (2, -3/(8)), (-1, -3/(-4)), etc. Didn't work.

    Here's a screen cap of the problem - http://i.imgur.com/k5XsDH6.png - if that can be of any help.
     
  5. Feb 24, 2013 #4

    SammyS

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    Of course, with (a, -3/(4a)), you must not include x = 0 , i.e. you can't have a = 0.

    Also, what is the second derivative if y = 0 , no matter the value of x ?
     
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