Math Is Fun Forum

mickeen

Member

Registered: 2009-04-18

Posts: 10

proving inequalities

Can someone give me a proof of the following inequality
(p+q)(q+r)(r+p) ≥ 8pqr

mickeen

nurshodiq

Member

Registered: 2009-03-29

Posts: 12

Re: proving inequalities

aritmetic mean

geometric mean

so

thanks

Last edited by nurshodiq (2009-04-24 01:24:17)

mickeen

Member

Registered: 2009-04-18

Posts: 10

Re: proving inequalities

Thanks for that but I dont really understand it. Can you five some explanation I can understand?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: proving inequalities

Hi Mickeen;

If p,q, and r are greater than 0 than there is a longer proof that only uses algebra and a little calculus.

Expand the left hand side:

(A)

subtract

from both sides

Divide both sides by

group the left hand side like this

Now look at the first bracket. It is the sum of a fraction and its reciprocal. We can think of it as

Try to prove that this function is always >= to 2. Take the derivative and set it to 0

solve the equation:

disregard the negative root and plug the 1 into

You get 2. Determine that x=1 is a minimum by the second derivative or by graphing the function. So now we know that the first bracketed term is >= 2. We can do the same thing for the remaining 2 bracketed terms. We have 3 terms all >= 2: The sum of them is >= 6. We have proven this:

This implies (A) for p,q and r >0

Last edited by bobbym (2009-04-28 04:12:43)

---

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.

mickeen

Member

Registered: 2009-04-18

Posts: 10

Re: proving inequalities

Bobbym, thanks for that! I can understand it better now.

mickeen

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: proving inequalities

Hi mickeen;

  Also the inequality is not true for negative numbers.

---

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.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: proving inequalities

That is a great proof, thanks bobbym, I learnt a lot from that.

However, is there a way to find the minimum of

without calculus?

Thanks also nurshodiq, I'm not that familiar with AMGM but it's interesting.

Last edited by Identity (2009-04-21 20:36:38)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: proving inequalities

Your welcome, glad it helped.

There are ways of getting that minimum without calculus but if I mention any of them I would be leaving myself wide open to much deserved criticism. There is a book by Ivan Niven that covers this particular subject.
Thanks for looking at the post.

See nurshodiqs good idea below!

Last edited by bobbym (2009-04-27 01:33:59)

---

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.

mickeen

Member

Registered: 2009-04-18

Posts: 10

Re: proving inequalities

Thanks very much for all of the above. Very good. Can someone give me  a proof of the following inequality please?
x^4 + y^4 + z^4 + w^4  ≥ 4xyzw

nurshodiq

Member

Registered: 2009-03-29

Posts: 12

Re: proving inequalities

we have aritmetic mean (AM) and Geometric mean (GM)

and

so for this problem we have

and both multiple by 4

thanks

Last edited by nurshodiq (2009-04-27 02:20:07)

nurshodiq

Member

Registered: 2009-03-29

Posts: 12

Re: proving inequalities

to find the minimum of

 

so minimum of

is 2

Last edited by nurshodiq (2009-04-27 00:44:16)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: proving inequalities

Good way to find the minimum without Calculus, Thanks

---

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.

mickeen

Member

Registered: 2009-04-18

Posts: 10

Re: proving inequalities

Nurshodiq, thanks for that

But how do we know that the
aritmetic mean ≥ geometric mean

or is there a proof of it?

Mickeen

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: proving inequalities

I think Wikipedia has a proof of the AM–GM inequality by mathematical induction.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: proving inequalities

Here it is:
http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means

---

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.