Math Is Fun Forum
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#1 2014-06-01 02:15:33
Inequality?
If a,b,c > 0
and abc=1
Prove that a^2 + b^2 + c^2 ≥ a+b+c
'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 2014-06-01 02:33:34
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Inequality?
Hi;
We have to assume that a ≠ b ≠ c else equality occurs at a=b=c=1?
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 2014-06-01 02:47:01
Re: Inequality?
Equality is allowed. See the geq sign in post 1. Please prove it now.
'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 2014-06-01 03:01:44
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Inequality?
That is one that I have and requires Cauchy Schwarz.
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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#5 2014-06-01 03:06:59
Re: Inequality?
The guy who will see it s one year older than me. He might know it. Please post it.
'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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#6 2014-06-01 03:11:19
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Inequality?
Let x = a, y = b, z = c:
Use CauchySchwarz to get:
By AM-GM and xyz=1 we have
So
using 1)
and so
For any real number t >= 1 we have t^2 >= t and t >= √t
so using 1)
we are done.
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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#7 2014-06-01 03:12:33
- ShivamS
- Member
- Registered: 2011-02-07
- Posts: 3,648
Re: Inequality?
I doubt he will know it, unless he has done some serious self-study.
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#8 2014-06-01 15:55:38
Re: Inequality?
He knew that inequality but I still do not. Can you tell me?
'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 2014-06-01 15:57:57
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Inequality?
Tell you what?
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 2014-06-01 16:07:10
Re: Inequality?
Cauchy Schwarz
'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 2014-06-01 16:28:23
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Inequality?
I have to go offline a bit. Will explain it when I get back.
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
#12 2014-06-01 17:23:58
Re: Inequality?
Okay
'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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#13 2014-06-01 18:26:34
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Inequality?
CS says
Now he just used b1=b2=b3 to get
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
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