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## #1 2012-07-24 01:03:47

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,824
Website

### Prove that

Prove that

without using the Geometric Mean Arithmetic Mean inequality

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #2 2012-07-24 02:45:55

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Prove that

Hi;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2012-07-25 01:17:10

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,824
Website

### Re: Prove that

What was that??!!

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #4 2012-07-25 01:28:25

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Prove that

I think you can use the inductive step of the inductive proof for the AM-GM inequality.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #5 2012-07-25 07:43:11

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Prove that

Hi Agnishom;

What is wrong?

The first line is from the problem itself. The second line is known to inequality provers. You can not even get it with a package! It is derived from this relationship:

Before we go further do you understand why the proof works? It is a very important concept in these type of proofs.

Would you like to see some more proofs of this?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #6 2012-07-26 00:48:41

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,824
Website

### Re: Prove that

No I really don't understand that.
I would be very happy to see it explained

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #7 2012-07-26 01:43:00

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Prove that

The first thing is that no constraints are given for the variables. We will assume that they are all >=0.

Look at the RHS. The first term is now clearly positive.

The second term is also positive because each term is squared.

Multiplying two positive numbers yields a positive number so we have:

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #8 2012-07-26 02:01:51

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,824
Website

### Re: Prove that

Hmmmm....
Thank You
I think my Father told me this but I forgot

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #9 2012-07-26 02:30:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Prove that

Hi;

It is a little easier to understand with a trivial example:

Prove, with a,b>=0;

Square both sides:

The boxed term is positive!

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #10 2012-07-26 04:36:17

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,824
Website

### Re: Prove that

My father showed me this one.
But he couldn't solve it for 3 variables

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #11 2012-07-26 05:25:23

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Prove that

The one you posted is quite difficult according to the book.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #12 2012-07-30 17:04:05

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,824
Website

### Re: Prove that

According to which book?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

## #13 2012-07-30 21:26:41

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Prove that

According to any book you used to teach pupils.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #14 2012-07-31 02:09:24

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Prove that

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #15 2012-08-01 02:21:37

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,824
Website

### Re: Prove that

May I ask why is my that topic on Geogebra closed?

"Of all the triangles possible within a givin perimeter, the equilateral triangle has the largest possible area"
What is this property called?
Am I allowed to call it "The Isoperimetric Property of an Equilateral Triangle"

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

## #16 2012-08-01 03:02:56

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Prove that

Hi;

Yes, you may ask that and it is a darn good question. I wanted to keep it clean until you were ready. It is now open for business.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline