Results 1 to 4 of 4

Math Help - Inverse and derivative

  1. #1
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Inverse and derivative

    Give an example of a differentiable function f: (0, \infty) \longrightarrow \mathbb{R} such that f '(x)=f^{-1}(x), \ \forall x > 0. Can you prove that there are no other examples?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Aug 2009
    Posts
    143
    Quote Originally Posted by NonCommAlg View Post
    Give an example of a differentiable function f: (0, \infty) \longrightarrow \mathbb{R} such that f '(x)=f^{-1}(x), \ \forall x > 0. Can you prove that there are no other examples?
    Here is one example:

    Let f(x)=ax^n, then
    f'(x)=anx^{n-1}
    f(f'(x))=f(anx^{n-1})=a(anx^{n-1})^n=a^{n+1}n^nx^{n(n-1)}=x

    Therefore
    n(n-1)=1, a^{n+1}n^n=1
    n_1=\frac{1+\sqrt{5}}{2}, a_1=\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{1-\sqrt{5}}{2}}
    n_2=\frac{1-\sqrt{5}}{2}, a_2=\left(\frac{1-\sqrt{5}}{2}\right)^{\frac{1+\sqrt{5}}{2}} (This one is not real).

    So,
    f(x)=\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{1-\sqrt{5}}{2}}x^{\frac{1+\sqrt{5}}{2}}
    Last edited by mr fantastic; September 19th 2009 at 01:38 AM. Reason: Restored original reply
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jan 2009
    Posts
    715
    Quote Originally Posted by luobo View Post
    Here is one example:

    Let f(x)=ax^n, then
    f'(x)=anx^{n-1}
    f(f'(x))=f(anx^{n-1})=a(anx^{n-1})^n=a^{n+1}n^nx^{n(n-1)}=x

    Therefore
    n(n-1)=1, a^{n+1}n^n=1
    n_1=\frac{1+\sqrt{5}}{2}, a_1=\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{1-\sqrt{5}}{2}}
    n_2=\frac{1-\sqrt{5}}{2}, a_2=\left(\frac{1-\sqrt{5}}{2}\right)^{\frac{1+\sqrt{5}}{2}} (This one is not real).

    So,
    f(x)=\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{1-\sqrt{5}}{2}}x^{\frac{1+\sqrt{5}}{2}}

    Is it the unique function for this ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by simplependulum View Post
    Is it the unique function for this ?
    yes! luobo's example, i.e. f(x)=\varphi^{1 - \varphi}x^{\varphi}, where \varphi=\frac{1+\sqrt{5}}{2}, is the only function that satisfies the condition. now the question is: why?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. derivative of inverse
    Posted in the Calculus Forum
    Replies: 9
    Last Post: March 20th 2010, 03:44 PM
  2. derivative of the inverse
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 28th 2010, 01:09 PM
  3. Inverse Derivative
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 21st 2009, 03:38 PM
  4. Derivative of Inverse
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 28th 2009, 01:00 AM
  5. plz help inverse derivative
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 8th 2006, 03:18 AM

Search Tags


/mathhelpforum @mathhelpforum