inequality

[chrislav]

Prove :

[MATH]\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq\frac{27}{2(a+b+c)^2}[/MATH]
Where a,b,c are positive Nos

 

[Steven G]

What have you tried? Where are you stuck? It is hard to help you if we do not know which method you are trying to use and where you are stuck. If you followed our guidelines you would have received help by now.

Also what is Nos???

 

[chrislav]

What have you tried? Where are you stuck? It is hard to help you if we do not know which method you are trying to use and where you are stuck. If you followed our guidelines you would have received help by now.

Also what is Nos???

I do not know how to start this inequality

 

[Steven G]

Prove :

[MATH]\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq\frac{27}{2(a+b+c)^2}[/MATH]
Where a,b,c are positive Nos

Maybe try getting common denominator? Try something!

 

[chrislav]

Maybe try getting common denominator? Try something!

thanks jomo i tried that but the problem seems to get more complicated

 

[Steven G]

thanks jomo i tried that but the problem seems to get more complicated

Can we please see your work? What did you use for a common denominator? BTW, I should have said to just get a common denominator for the lhs.

 

[chrislav]

before we continiue do you know the proof of that inequality

 

[pka]

before we continiue do you know the proof of that inequality

Actually I have chosen not to even work on this because I doubt that it is true.
I say that because I have never see any problem even close to this one.
I will suggest a nice book that may help: An Introduction to Inequalities by Beckenbach & Bellman.

 

[Steven G]

before we continiue do you know the proof of that inequality

No

 

[lookagain]

Actually I have chosen not to even work on this because I doubt that it is true.
I say that because I have never see any problem even close to this one.

Thank goodness I'm not a "true PhD mathematician" to be dismissive of such a problem. Rather, I knew to have an option
to plow ahead and test a = b = c, for starters. That comes up with the inequality \(\displaystyle \ 3/2 \ \ge \ 3/2. \ \ \) That's promising.

Then, I may look into letting b = a, and c = ka, where 0 < k < 1 for one case and k > 1 for the other case
(so that I have two variables equal to each other, and the third variable is distinct from them).

From that I got it down to \(\displaystyle \ (2 + k)^3 \ \ versus \ \ 27k. \ \ \) Someone may plot \(\displaystyle \ y = (2 + x)^3 \ \ and \ \ y = 27x\)
on the same set of axes to help see the cases mentioned above.

This continues to look promising.

But then there is the \(\displaystyle \ a \ne b \ne c \ \) case still to work with . . .


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I am supposing that this inequality, if true, was not proved by cases as I am hinting at.

 

[topsquark]

before we continiue do you know the proof of that inequality

Well, that was insulting. Thank you.

-Dan

 

[chrislav]

Well, that was insulting. Thank you.

-Dan

I am sorry if offended anyone

 

[topsquark]

I am sorry if offended anyone

I was feeling a little ? yesterday. My apologies in turn.

-Dan

 

[chrislav]

o.k Dan ,thank you

 

[Steven G]

So back to the problem....
Let's see the results with a common denominator for the lhs. I can't imaging getting anywhere without seeing this.

 

[Dr.Peterson]

Prove :

[MATH]\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq\frac{27}{2(a+b+c)^2}[/MATH]
Where a,b,c are positive Nos

You haven't stated the context of the problem, which can affect what methods you might be expected to use. Would you be able to use partial derivatives to find the minimum of the difference between the sides? Or have you been learning some particular techniques for proving inequalities?

 

[chrislav]

Or have you been learning some particular techniques for proving inequalities?

No,can you suggest some

 

[Dr.Peterson]

The intent of the question was to find out what you know that you can use. Please answer the other questions; observe our guidelines as stated here, which ask for context and work.

Is this a contest problem, or homework for a course, or something else? What other mathematics have you learned?

 

[Steven G]

I always thought that it was a contest problem but can be wrong.

 

[chrislav]

I always thought that it was a contest problem but can be wrong.

you are right this is a contest problem.

I thing my mathematics are enough for that enequality