Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

I am useless at proving inequalities

Kurtz

Member

Registered: 2007-10-01

Posts: 27

Re: I am useless at proving inequalities

I hope I'm not spamming because I have alot to learn still....

But anyways have you tried

if abc= 1 then   1 + 3     ≥     6
                            ___        ____
                         Variable    Double Variable

I'm thinking that A+B+C = 1:D plz dont get mad if it seems off the wall.
and it may go the same with the other one but with two abc variable but mixed up which adds to 2, so it might look like this.....

1+ 3  ≥   6
    ---     ---    =    4  ≥  6
     1       2           ---   ---  =   4 ≥ 3              to me it seems to prove the problem hope i helped
                            1     2

Last edited by Kurtz (2008-08-08 14:07:29)

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I am a mathemagician. You ask why? I point up

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: I am useless at proving inequalities

I'm thinking that A+B+C = 1

a, b, and c must all be positive.  Thus, if a+b+c = 1, then they must all be less than 1 and greater than 0.  However, then it has to be a*b*c < 1, which contradicts the fact that a*b*c = 1.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Kurtz

Member

Registered: 2007-10-01

Posts: 27

Re: I am useless at proving inequalities

So what if A*B*C=1 then wouldnt it also all be 1?
because A times B times C = 1 which could be substituted as A=1 B=1 C=1

which is A(1)*B(1)*C(1)=1

Then we're proving the equation false?

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I am a mathemagician. You ask why? I point up

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: I am useless at proving inequalities

Kurtz, try A = 1/2, B = 2, C = 1

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: I am useless at proving inequalities

It appears with number fiddling that:
(ab+bc+ca-3)*2 > a+b+c-3 when ab+bc+ca < 6 anyway.
For ab+bc+ca > 6, it doesn't matter anymore.

Last edited by John E. Franklin (2008-08-08 18:06:28)

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igloo myrtilles fourmis

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: I am useless at proving inequalities

http://i41.photobucket.com/albums/e271/ … ngHead.gif

perhaps the solution lies in the relationship with those equations in a,b,c and a cubic polynomial:

a b c are roots of the cubic equation x³ + Ax² + Bx + C = 0 such that.

abc = -A = 1
ab+bc+ac = B
a+b+c = -C

also ofcourse; x³ + Ax + Bx + C = (x-a)(x-b)(x-c)
and so far we have: x³ - x² + Bx + C = 0

since a,b,c are positive, C must be negative, and B must be positive, hence;

well now i'm stuck and tired and going to bed ;P

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: I am useless at proving inequalities

a b c are roots of the cubic equation x³ + Ax² + Bx + C = 0 such that.

abc = -A = 1
ab+bc+ac = B
a+b+c = -C

You got the A and the C in the wrong places.

Anyway, while floundering in the sea of despair, I stumbled upon another inequality:

My attempt was as follows. If

, we have equality. Otherwise, given that

, we must have either

or

.

I can prove the inequality for the case

.

But I can’t do it for the case

.

Ooh, I was sooo close!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: I am useless at proving inequalities

Hopefully this works.

Let

. We have

.

So the inequality is

By Cauchy–Schwarz

.

Hence, we have

and we are done if we can show that

.

Well, writing

, we have

and we are indeed done.

ZHero

Real Member

Registered: 2008-06-08

Posts: 1,889

Re: I am useless at proving inequalities

Who said that you're USELESS......?

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Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: I am useless at proving inequalities

uhm didnt work out

(why do sometimes this forum post when i click preview??)

Last edited by Kurre (2008-08-09 21:47:19)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: I am useless at proving inequalities

What didn’t work out?

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: I am useless at proving inequalities

Well I was writing a solution and when I clicked preview It got posted instead and then I didnt manage to complete the proof. I substituted c=1/ab and the inequality for your case reduced to:

for a,b≤1

Last edited by Kurre (2008-08-09 23:58:59)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: I am useless at proving inequalities

I tried that too. In fact,

and

– so I tried using the result I had already proved:

It was no good.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: I am useless at proving inequalities

tony123 strikes.

http://www.mathlinks.ro/viewtopic.php?p=1219810#1219810

tony123

Member

Registered: 2007-08-03

Posts: 229

Re: I am useless at proving inequalities

Set

. With some calculations, we get the new problem accompanying the condition

By  AM-GM, we have the simple inequality:

in order to imply the important result

( which is what we want! )

tony123

Member

Registered: 2007-08-03

Posts: 229

Re: I am useless at proving inequalities

The inequality is equivalent to

By AM-GM Inequality,

.

It is obvious that

,

so we are done!

Last edited by tony123 (2008-08-11 00:06:43)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: I am useless at proving inequalities

The inequality is equivalent to

By AM-GM Inequality,

.

It is obvious that

,

so we are done!

Thank you!

MooseofDoom

Member

Registered: 2008-08-10

Posts: 13

Re: I am useless at proving inequalities

oh I though a,b,c were = 1 so that:

1+3/((1)+(1)+(1)) 1+6/((1*1)+(1*1)+(1*1))

1+1=2 1+2=3.  Oh but then 2 is not equal to 3

Hmm... So who figured it out?

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Kurtz

Member

Registered: 2007-10-01

Posts: 27

Re: I am useless at proving inequalities

Moose that thinking process has been done. I believe I post it above but also another thing is that (A)(B)(C) can not all equal 1 and like Identity showed me is that we must use invert substitution that will also give us the result of one. Not just 1*1*1.

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I am a mathemagician. You ask why? I point up