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  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

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#1 2012-10-26 20:46:42

Harold
Guest

Inequality with pi and e

What is the solution of this problem?and how is the problem solved-

#2 2012-10-26 21:26:19

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,366

Re: Inequality with pi and e

Hi Harold;


This one has been around for a long time. The standard answer starts with raising both sides to the power of

after that it is a maxima-minima problem.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2012-10-26 21:47:54

Harold
Guest

Re: Inequality with pi and e

You mean e^e is always bigger than pi^pi?but why?

#4 2012-10-26 21:50:54

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,366

Re: Inequality with pi and e

Hi;

e^e is not greater than π^π. That is not what I said. You did something wrong with the first step.

From here it is an ugly calculus problem.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2012-10-26 22:31:57

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,380

Re: Inequality with pi and e

Looks to me that y = x^(1/x) has a single maximum at x = e.

See graph and derivative graph.

Bob

View Image: Harold.gif

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#6 2012-10-27 11:47:28

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: Inequality with pi and e

Let
; then
when
. So
is decreasing for
; as
,
, i.e.
.

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