My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

Thanks Tree4Thanks
  • 2 Post By mathman
  • 2 Post By romsek
LinkBack Thread Tools Display Modes
November 4th, 2017, 11:05 PM   #1
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
$\displaystyle a^a$ = $\displaystyle b^b$.
Attached Images
File Type: jpg Q1AS7.jpg (7.5 KB, 14 views)
yli is offline  
November 5th, 2017, 01:11 PM   #2
Global Moderator
Joined: May 2007

Posts: 6,378
Thanks: 542

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.
Thanks from topsquark and yli
mathman is offline  
November 5th, 2017, 01:38 PM   #3
Senior Member
romsek's Avatar
Joined: Sep 2015
From: Southern California, USA

Posts: 1,602
Thanks: 816

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
romsek is offline  

  My Math Forum > College Math Forum > Calculus

interpreting, problem, proving

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Interpreting a statement f(x,y) EXP Linear Algebra 4 March 8th, 2016 06:21 AM
Interpreting results doryyroryy Calculus 4 July 12th, 2015 04:26 PM
Box and whiskers diagram - help interpreting it please matheus Probability and Statistics 7 April 19th, 2015 08:41 AM
Error in interpreting differentiation. jim198810 Real Analysis 6 April 11th, 2015 01:33 PM
Please help interpreting this graph. techdaemon Advanced Statistics 1 February 28th, 2010 11:17 PM

Copyright © 2017 My Math Forum. All rights reserved.