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Help needed: proof of an increasing function

  1. Jan 12, 2013 #1
    I am supposed to show f(x) is increasing in x when x is real and positive.

    f(x)= [(x+2)*(1-a^(x+1)] / [1-(x+2)*(1-a)*a^(x+1)-a^(x+2)]

    a is any real in (0,1); x is real and positive

    I have taken and first derivative of f(x):

    f'(x)=1-a^(x+1)-a^(x+2)+(x+1)*(x+2)*log(a)*(a^(x+1)-a^(x+2))+a^(2x+3)

    The problem is I cannot compare log(a) with the power of a. Can any of you genius help me with a proof as to showing f'(x) >0? Or maybe there is some other way to show f(x) is increasing in positive x?
     
    Last edited: Jan 12, 2013
  2. jcsd
  3. Jan 12, 2013 #2

    haruspex

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    Please clarify the bracketing, preferably with LaTex. Is it
    ##f(x)=\frac{(x+2)*(1-a^{x+1})}{1-(x+2)*(1-a)*a^{x+1}-a^{x+2}}## ?
     
  4. Jan 12, 2013 #3
    Yes, it is. I am sorry about the typing/.

    f(x)=[itex]\frac{(x+2)*(1-a^{x+1})}{1-(x+2)*(1-a)*a^{x+1}-a^{x+2}}[/itex]
     
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