xth root of x
Just how big can x^(1/x) be? Assume x is real number.
Labels: mathschallenge, SharedPuzzle
posted by Chris at
1:19 PM
Monday, August 31, 2009
Links
Search
Share
- Tom Chrome Extension
- ToM @ Twitter
- MeeT ToM
- Daily Tricks Widget
- Vista Gadget 1.5
- iPhone Widget
- Windows XP Widget
- Google Toolbar Button
- Suggestions
Favorite Links
- Vista Sidebar Gadgets
- Widgets for Web 2.0
- C Sharp Tricks
- Interview Information
- Illustrator - Elar Alexander
Bookmark Us
Previous Posts
- NEVER is prime
- Who squares?
- Just making a point
- Happy Birthday
- Treasure Revisited
- Needle Mileage
- Treasure hunt
- Family Ties
- Miller's Fee
- Bug on a Rubber Band
12 Comments:
Max value of x is e (i.e. 2.71828183...)
Roll the drums ... we have a winner! First post too.
well, i supposed something like that, but until now i came up to:
d/dx[x^(1/x)*(1/(x^2)+log(x))]
still need the root...
Another good question Chris
I am not in any way a mathematician(did A level pure maths and just scraped through)
I just about manage to understand the answer e.
one question from me.... what happens when the value of x approaches infinity?
I'm not trying to outfox you, I only wondered.
Hi t::b:H (you're gonna be tbH when I answer, I get spots in front of my eyes reading your isospin name;)
I've lost my notes, but can see you've gone wrong there. I converted to e^((ln(x)/x) first. The rest's a doddle.
I don't want to give too much away. If you don't know calculus, you'll find it a tough nut to crack. Same goes for part 2 of "Hold on".
Unless there seems to be no point, I always publish a (over)full answer. Although if I'm lazy, I'll simply provide a link.
... I realised that it was a genuine question. There no site rule to say you can't try to outfox me though ;)
I would have been curious too. But I've seen the graph. Here you go:
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
About half way down.
Time's up.
Let y = x^(1/x) = e^(ln(x)/x)
dy/dx = y' = e^(ln(x)/x)(1/(x^2)-ln(x)/(x^2))
= e^(ln(x)/x)(1-ln(x))/(x^2) [don't you love in-line maths!]
For a critical point need y' = 0 => 1 = ln(x) => x = e
So the maximum value is e^(1/e) = 1.44466786...
So, silly old me, has incorrectly announced a winner in error.
I asked for max value of x^(1/x), not for the corresponding x.
However, that's nit-picking. So I'll let the winner list stand.
The ever present e has reminded me of a lecture I went to which include a reasonable argument, based on cost, that the best number of logical states for a computer to work with is e. So 2 or 3 is nearest. I notice that most logic circuits are tri-state => they can disconnect themselves from the circuit as well as putting out a 0 or 1.
hmmmrmrmramsrsar....
thanks, got involved in a chainrule derivation war... that e^ln is for sure the better solution =)
Hi t:b:H. I can't even remember d/dx(a^x) = ln(a)*a^x. I always do a^x = e^(x ln(a)) first. I've not had to this stuff in real life for about 40 years and that was only to get a degree (in electronics). But it's a bit like riding a bike.
Hi there! e as the maxima of the x root of x is one of my favorite math items. I learned of it from e: the story of a number. i just used the euler program to confirm this. strange. e is some kind of strange fulcrum.
Post a Comment
Links to this post:
Create a Link
<< Home