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Prove a composite function is increasing

  1. May 9, 2010 #1
    1. The problem statement, all variables and given/known data
    Hi,
    I have trouble proving this claim and would really appreciate your help =). Thank you in advance!
    So here's the question: Suppose that f is a continuous function for all x>= 0 and differentiable for all x> 0. Also, f(0) = 0 and f' (1st derivative of f) is increasing on its domain Define
    g(x) = f(x)/ x for x >0.
    Show that g is increasing for all x >0.

    3. The attempt at a solution
    I used the Quotion Rule to write g' = [xf'(x) - f(x)]/ x^2.
    For g to be increasing, g' must be >= 0 --> the question becomes: proving xf'(x) - f(x) >= 0.
    Since f' is increasing -> f'(x) > = f(0) with x >0
    --> It is sufficient to have xf'(0) - f(x) >0
    I wrote f'(0): by definition of derivative:
    f'(0) = lim [f(x) - f(0)]/ x-0 for x --> 0
    = lim f(x)/x
    Here I got stuck with the limit; it does not allow me to cross-multiple the terms.
     
  2. jcsd
  3. May 10, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    so rearranging it remains to show that
    [tex] f'(x) > \frac{f(x)}{x}[/tex]

    continuous & differentiable work with the mean value theorem, so there exists 0<c<x such that
    [tex]f(c) = \frac{f(x) -f(0)}{x-0}[/tex]
     
  4. May 10, 2010 #3
    Thanks for answering my question =). I couldn't believe that I didn't think about the Mean Value Theorem...
     
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