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Determining the bigger number

  • Thread starter Saitama
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  • #1
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Homework Statement


Use the function ##f(x)=x^{1/x} \, ,\, x>0##, to determine the bigger of two numbers ##e^{\pi}## and ##\pi^e##.


Homework Equations





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!
 

Answers and Replies

  • #2
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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)?
 
  • #3
haruspex
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. The function increases when x>1/e and decreases when 0<x<1/e.
Doesn't sound right. What do you get for x = 1, 2, 4, 16?
 
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  • #4
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Doesn't sound right. What do you get for x = 1, 2, 4, 16?
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?
 
  • #5
Dick
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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?
Looks correct.
 
  • #6
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Looks correct.
Thank you Dick and haruspex! :smile:
 

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