1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Help with functional equation

  1. May 16, 2012 #1
    1. The problem statement, all variables and given/known data
    A function from R-->R is differentiable and follows f( (x+y)/3 ) =( 2 + f(x) + f(y) ) / 3
    Derivative of f(x) at x=2 is 2

    Find the range of f ( |x| )


    2. Relevant equations



    3. The attempt at a solution
    Well the questions asks me the range of f( |x| ). But i don't even know f(x). I did find the value of f(0) which is 2. And i don't know what to do afterwards. I have no idea why the value of derivative is given.
     
  2. jcsd
  3. May 16, 2012 #2

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    How is f(-x) related to f(x) ?
    Hint: Let y = -x .
    Use that result to see how f '(-x) is related to f '(x) .
     
  4. May 16, 2012 #3

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Two more substitutions that might be helpful:
    y = x (which you may have used to fin f(0) )

    and

    y = 2x .
     
  5. May 17, 2012 #4
    Ok (by using spoiler) f(0) = 2 + f(x) + f(-x) / 3
    => 6 = 2 + f(x) +f(-x)
    => 4 = f(x) +f(-x)
    Thus 0 = f '(x) - f '(-x)
    f '(x) = f '(-x)
    f '(x) is even so f(x) is odd.

    But if f(x) is odd then f(x) + f (-x) = 0 but i found out earlier that it is equal to 4. A bit confused
     
  6. May 17, 2012 #5

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    If f(x) is odd, then it is true that f '(x) is even, but the converse is not necessarily true.

    If f '(x) is odd, then it does follow that f(x) is even.

    In your case, 4 = f(x) +f(-x), so that f(-x) = -f(x) + 4 . Therefore, f(x) is not odd.

    Consider the function g(x) defined as g(x) = f(x) - 2.

    g(x) is odd.
     
  7. May 17, 2012 #6
    Oh ok - but still i cannot solve the initial problem. This is what i have done till now:-
    f(0) = 2
    f(x) = f(-x) + 4 and f '(x) = f ' (-x)
    Are they right? and what else do i need to do?
     
    Last edited: May 17, 2012
  8. May 17, 2012 #7

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    See what derivative relationships result from the above.

    Also, you might consider investigating the behavior of the function g(x) .
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook