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Determining the bigger number

  1. Sep 17, 2013 #1
    1. The problem statement, all variables and given/known data
    Use the function ##f(x)=x^{1/x} \, ,\, x>0##, to determine the bigger of two numbers ##e^{\pi}## and ##\pi^e##.


    2. Relevant equations



    3. The attempt at a solution
    I honestly don't know where to begin with this problem. I found the derivative but that seems to be of no help. The function increases when x>1/e and decreases when 0<x<1/e.

    Any help is appreciated. Thanks!
     
  2. jcsd
  3. Sep 17, 2013 #2

    mfb

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    Hint: a<b is equivalent to a^c < b^c for positive a,b,c. Can you find a c such that your expressions look like x^(1/x)?
     
  4. Sep 17, 2013 #3

    haruspex

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    Doesn't sound right. What do you get for x = 1, 2, 4, 16?
     
  5. Sep 17, 2013 #4
    Oops, its just the opposite of what I posted.

    I have tried the problem again.
    ##f(e)>f(\pi) \Rightarrow e^{1/e}>\pi^{1/\pi}##
    Raising both the sides to the power ##e\pi##
    $$e^{\pi}>\pi^e$$

    Looks correct?
     
  6. Sep 17, 2013 #5

    Dick

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    Looks correct.
     
  7. Sep 17, 2013 #6
    Thank you Dick and haruspex! :smile:
     
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