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2 Limits/Differenciability questions

  1. Sep 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Q1. let g(x) = log(f(x)), where f(x) is a twice diffrenciable positive function on (0, inf) such that f(1+x) = xf(x)

    Then for N = 1,2,3.....

    g''(N+1/2) - g''(1/2) = ??

    Q2. Let f(x) be differenciable on the interval (0, inf) such that f(1) = 1, and Lim(t-->x) [t^2f(x)-x^2f(t)]/t-x = 1 for each x > 0

    Then f(x) is...??

    2. Relevant equations
    none I believe....?

    3. The attempt at a solution

    I tried Q1 by finding g''(x) and f''(x) and then putting them into the raw equation, using the given condition f(1+x) = xf(x) and writing
    g''(1/2) as g''(-1/2+1)
    and g''(N+1/2) as (n-1/2 +1)

    But to no avail

    P.S. The problems are from a MCQ test......tell me if you'd need the options aswell....I'll be happy to provide them! :)
  2. jcsd
  3. Sep 10, 2009 #2


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    I assume you mean with parentheses (t-x) in the denominator.
    Try adding and subtracting x^2f(x) in that numerator, should lead you to a first order linear DE.
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