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November 4th, 2017, 10:05 PM  #1 
Newbie Joined: Oct 2017 From: ... Posts: 5 Thanks: 0  Need help interpreting and proving this problem.
I'm unsure about how to approach this question. It is confusing me because b is supposed to be larger than a, but the example has two values that equal each other. Any help is appreciated. In case the picture is difficult to read I have typed the question out as well: Note that: $\displaystyle (\frac{1}{2})^\frac{1}{2}=(\frac{1}{4})^\frac{1}{4 }$ Explain why there are infinitely many pairs of numbers a < b such that $\displaystyle a^a$ = $\displaystyle b^b$. 
November 5th, 2017, 12:11 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,454 Thanks: 567 
If you look at the curve for 0<x<1, you will see a curve with a minimum at x=1/e, so that there are pairs of x's where the function has the same value.

November 5th, 2017, 12:38 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 1,791 Thanks: 923 
as they suggest consider $y=x^x$ This function has a minimum of $e^{1/e}$ at $x = \dfrac 1 e$ and it rises to $(0,1)$ to the left, and off to infinity to the right. So you can set a horizontal line $y = y_0,~y_0 \in \left(\dfrac 1 e,~1\right]$ and intersect $y=x^x$ in two places, one, $a$, to the left of $\dfrac 1 e$, and $b$, to the right of it. and from this $a^a = b^b = y_0$ clearly there are an infinite number of $y_0 \in \left(\dfrac 1 e, ~1\right]$ and thus an infinite number of $(a,b)$ pairs. 

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interpreting, problem, proving 
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