My Math Forum Need help interpreting and proving this problem.

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November 4th, 2017, 10:05 PM   #1
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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
$\displaystyle a^a$ = $\displaystyle b^b$.
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 November 5th, 2017, 12:11 PM #2 Global Moderator   Joined: May 2007 Posts: 6,607 Thanks: 616 If you look at the curve for 0
 November 5th, 2017, 12:38 PM #3 Senior Member     Joined: Sep 2015 From: USA Posts: 2,120 Thanks: 1101 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. Thanks from topsquark and yli

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