Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Tough Inequality

Hi;

This one is pretty tough but see what you can do.

For positive numbers a,b,c:

Last edited by bobbym (2009-09-03 09:27:24)

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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.

rzaidan

Member

Registered: 2009-08-13

Posts: 59

Re: Tough Inequality

Hi bobbym
I am online now just to say hi
Riad Zaidan

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Tough Inequality

Last edited by JaneFairfax (2009-09-03 22:11:32)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tough Inequality

Hi Jane;

Nice solution!

Last edited by bobbym (2009-09-04 15:21:51)

---

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.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Tough Inequality

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Tough Inequality

[align=center]

[/align]

Nobody?

Please give it a try. It’s actually very simple.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tough Inequality

Please give it a try. It’s actually very simple.

Please forgive me Jane, I have been working on it. I think it is one you give to your worst enemy.

Last edited by bobbym (2009-09-11 06:56:00)

---

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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tough Inequality

Hi Jane;

I couldn't get it, can you show me your solution. You know, the simple one. I have exhausted all of my ideas.

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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.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Tough Inequality

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Why did the vector cross the road?
It wanted to be normal.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tough Inequality

Hi mathsyperson;

I avoided the idea of minima because I don't think that is the way Jane went. Now after lots of dead ends your hint is my only chance.

Prove:

Minimizing the first radical of A:

Minimum at b = 2a

Minimizing the second radical of A:

minimum at b = 2c

Plugging a = b / 2 and c = b /2 into:

We get:

Squaring both sides to make use of the fact that:

Since the 3 b^2 on the left represents the minimum for the LHS of A)  we have proved:

Last edited by bobbym (2009-09-12 17:20:55)

---

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.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Tough Inequality

If you want to avoid calculus, you could also say

Clearly the squared part is non-negative and so the minimal value must be -b²/4.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Tough Inequality

Last edited by JaneFairfax (2009-09-13 14:38:23)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tough Inequality

Hi Jane;

Thanks for your method and thanks mathsy for yours.

Last edited by bobbym (2009-09-13 14:27:15)

---

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.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Tough Inequality

A problem similar to mine can be found here.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tough Inequality

Hi Jane;

Thanks so much for the link. I appreciate that. Hmmm, that ValentineA again ?!

Last edited by bobbym (2009-09-15 08:08:17)

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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.