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October 18th, 2018, 01:38 PM   #1
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Inequality tips

Hi, so I'm working on an inequality problem, and the problem gives you that $a,b,c$ are real numbers and $a+b+c=3$. Using this and Cauchy Schwarz, I was able to get that $a^2+b^2+c^2 \ge 3$ and $-3 \le ab+bc+ca \le 3$. This helps out, but there is still $abc$ I would like to know something about, but can't seem to figure out any bounds on it.

Does anyone have any ideas? Specifically there is a term $+18abc$ which I'd like to be $\ge 0$ and I've written that as $2(a+b+c)^2(abc)$ but we obviously can't conclude that $abc$ is \ge0 since they are real numbers.

Sorry if this is confusing. I'm mainly looking for ideas on what I could do.
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October 18th, 2018, 02:25 PM   #2
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It seems you are leaving out some information. You can't conclude anything about the sign of $abc$ for exactly the reasons you have stated.
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October 18th, 2018, 02:39 PM   #3
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Quote:
Originally Posted by SDK View Post
It seems you are leaving out some information. You can't conclude anything about the sign of $abc$ for exactly the reasons you have stated.
I can just figure it out, thanks!
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October 18th, 2018, 04:14 PM   #4
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There are no bounds to $abc$. For instance, with a large positive $bound$ take the following values:

$a = +bound$
$b = -bound+4$
$c = -1$

The special case is where $a, b, c$ are all positive real numbers. Not hard to see that $0 <= abc <= 1$

Last edited by Arisktotle; October 18th, 2018 at 04:17 PM.
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October 18th, 2018, 04:23 PM   #5
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Originally Posted by Arisktotle View Post
There are no bounds to $abc$. For instance, with a large positive $bound$ take the following values:

$a = +bound$
$b = -bound+4$
$c = -1$

The special case is where $a, b, c$ are all positive real numbers. Not hard to see that $0 <= abc <= 1$
Okay, thank you! I will try to incorporate this term into the rest of the inequality. I appreciate it!
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October 19th, 2018, 11:40 AM   #6
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This is my 5th inequality proof and it's crazy how many tricks of the trade I learned, I had to completely change my approach!

Lesson: if an approach isn't working, don't keep trying it
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