1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

A analysis problem

  1. Feb 20, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]f[/tex] is continuous on [a,b]
    [tex]f_{1}(x)=\int^x_a f(t)dt[/tex]
    [tex]f_{2}(x)=\int^x_a f_{1}(t)dt [/tex]
    [tex]\forall x\in[a,b],\exists n[/tex] depends on x , such that [tex]f_{n}(x)=0.[/tex]
    prove that [tex]f\equiv0.[/tex]

    2. Relevant equations
    mathematical analysis
    3. The attempt at a solution
    copy the taylor theorem 's proof???
    but I get nothing.
  2. jcsd
  3. Feb 21, 2010 #2


    User Avatar
    Science Advisor

    If, for some n, fn is identically 0, then it is a constant and so its derivative is identically equal to 0. But, by the fundamental theorem, the derivative of [itex]f_n(x)= \int_a^x f_{n-1}(t)dt[/itex] is fn-1(x). Therefore, if fn(x) is identically 0 on [a, b], so is fn-1(x).
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook