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Functional Equation (Probably needs a CAS)

  1. Jun 1, 2009 #1
    1. The problem statement, all variables and given/known data
    If f'(x)>0 for all real positive x, where f:R+ ---> R and
    f(x)+(1/x)=f-1(1/(f(x))),

    f-1(1/(f(x)))>0 for all x>0. Find all the possible values of (i) f(2),(ii) f'(2) and (iii) Limit (x f(x)) as x ----->0 .


    3. The attempt at a solution
    Guessing from the last part, I intuitively, took f(x) = k/x. Using this, I get a quadratic in k which yields that k might be [tex]\frac{1 (+/-) \sqrt{5} }{2}[/tex]. But this was just a guess. And there might still be other functions which satisfy that equation. I wonder if they could be manually solved for.
    All that I need from you guys is to check out if all the possible solutions of this equation can be solved manually or not. If not, please let me know how can I get solutions to this equation using a Computer Algebra System. Actually, I have both Matlab R2008a and Mathematica 7 but I dont know how to use them to solve functional equations involving inverses. If anyone of you could provide me the syntax for these CAS on how to solve functional equations, I shall be very grateful.
    Thanks.
    Ritwik
     
  2. jcsd
  3. Jun 5, 2009 #2

    tiny-tim

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    Homework Helper

    Hi Ritwik! Thanks for the PM! :smile:

    Yes, f(x) = (1 ± √5)/2x certainly works …

    I think you've done very well to get that!

    Sorry, but I haven't a clue how to get any other solution :redface:

    (and I don't know any computing)

     
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