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!

Derivative/Integral proof

  1. Jan 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that if f(x) is continuous and [tex]f(x) = \int_0^x f(x) dx[/tex], then f(x) = 0.

    2. Relevant equations



    3. The attempt at a solution

    If [tex]f(x) = \int_0^x f(x) dx[/tex], then by integrating by the FTC we have f'(x) = f(x). Thus the only solution to this equation will have the form [tex]f(x) = ce^x[/tex] for some constant c. Now, [tex]f(x) = \int_0^x f(x) dx = f(x) - f(0) [/tex], implying that f(0 = 0. So since we know the solution to the equation will be [tex] f(x) = ce^x[/tex] then we have [tex] 0 = f(0) = ce^0 = c[/tex], implying that c = 0. Thus f(x) = 0. QED

    Is this correct?
     
  2. jcsd
  3. Jan 29, 2009 #2

    Mark44

    Staff: Mentor

    I'm confused by two things:
    1. Your use of x in one of the limits of integration and as the dummy variable in the integral. It would be better to use different variables.
    2. Your use of ' (as in dx'). Is this supposed to mean the derivative with respect to x of the definite integral?
    Based on these points, I believe you are saying that
    [tex]f(x) = \frac{d}{dx}\int_0^x f(t)dt[/tex]

    Now maybe I've missed something in how I've interpreted your problem, but the equation just above is true for every function f that is continuous on [0, b], and where 0 <= x <=b, per the FTC, so it does not follow that f(x) is identically 0.
     
  4. Jan 29, 2009 #3
    Oops, let me retype the question:

    Prove if f(x) is continuous and [tex]f(x) = \int_0^x f(x) dx[/tex] then f(x) = 0.

    That's the question, and the mark beside the dx was only a comma, it wasn't meant to denote the derivative of the integral. So now that I fixed the question, isn't f'(x) = f(x)? And so f(x) = ce^x, but 0 f(0) = ce^0 = c and so c = 0 and f(x) = 0.
     
  5. Jan 29, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You still may be causing some confusion by having the dummy variable of integration be the same as the limit, but I get what you are saying. I think that proof is ok. As f is the integral of a continuous function it's differentiable.
     
  6. Jan 29, 2009 #5

    Mark44

    Staff: Mentor

    Same here.
     
  7. Jan 29, 2009 #6
    Ugh, I meant to change the variable of integration to t but I forgot! Sorry, and thanks for checking my work.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Derivative/Integral proof
  1. Derivative proof (Replies: 4)

  2. Derivative Proof (Replies: 6)

  3. Derivative Proof (Replies: 11)

Loading...