Results 1 to 2 of 2

Math Help - Weird Induction proof

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    5

    Weird Induction proof

    Suppose f:R-->R satisfies f(xy) = xf(y)+yf(x) for all real numbers x,y. Prove that f(1)=0 and that f(u^n)=n*u^(n-1)*f(u) for all natural numbers n and all real numbers u.

    I really have no clue where to start on this proof, except that the hint in the problem says that in using induction on n, consider the case u = 0 separately.

    Any help would be much appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by morbius27 View Post
    Suppose f:R-->R satisfies f(xy) = xf(y)+yf(x) for all real numbers x,y. Prove that f(1)=0 and that f(u^n)=n*u^(n-1)*f(u) for all natural numbers n and all real numbers u.

    I really have no clue where to start on this proof, except that the hint in the problem says that in using induction on n, consider the case u = 0 separately.

    Any help would be much appreciated.
    For the first, let y=1 then f(x)=xf(1)+f(x) and so xf(1)=0 and since x was arbitrary, it follows that f(1)=0. For the second suppose you know it for n then f(u^{n+1})=f(u^nu)=u^nf(u)+uf(u^n)=u^nf(u)+unu^{n-1}f(u)=(n+1)u^nf(u)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Weird (difficult) proof
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: March 24th 2009, 09:21 PM
  2. Mathemtical Induction Proof (Stuck on induction)
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: March 8th 2009, 09:33 PM
  3. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: June 8th 2008, 01:20 PM
  4. [SOLVED] A weird limit proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 24th 2008, 12:41 PM
  5. weird log proof
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 26th 2006, 07:21 AM

Search Tags


/mathhelpforum @mathhelpforum