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Proving convexity

  1. Sep 1, 2008 #1
    1. The problem statement, all variables and given/known data

    f(x,y) = X[tex]\alpha[/tex]Y[tex]\beta[/tex]

    (that's X to the power of alpha, Y to the power of Beta)

    Is this function convex? Prove it.

    2. Relevant equations

    f''(x,y) > 0 ==>convexity

    3. The attempt at a solution

    My steps are as follows:

    f(X,Y) = (X^a)(Y^ß)

    f’(X,Y) = (aX^a-1)(Yb) + (X^a)(bY^b-1)

    f’’(X,Y) = ((a^(2)-a)X^a-2)(Yb) + (aX^a-1)(bY^b-1) + (aX^a-1)(bY^b-1) + (X^a)((b^(2)-b)Y^b-2)

    I'm just trying to simplify it to prove it is greater than 0 (or not). Is my work correct so far and how can I be sure the second derivative is indeed positive? Thanks a lot.
  2. jcsd
  3. Sep 2, 2008 #2


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    Hi 3.141592654! :smile:

    (have an alpha: α and a beta: β :smile:
    … and look above the reply box for the X2 tag :wink:)

    What is your ' differentiating with respect to? :confused:

    You have a 3-D surface, with two variables. :wink:
  4. Feb 10, 2009 #3
    Is it possible to use the definition of convexity to show a function is convex? Or do you just have to show the 2nd derivative is positive for all x?

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