Solve Inequalities with Traditional x^2 or x^4

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Discussion Overview

The discussion revolves around solving a specific inequality involving four positive variables \(a\), \(b\), \(c\), and \(d\) under the constraint that their product equals one. Participants explore methods to demonstrate that the expression \(a^2 + b^2 + c^2 + d^2 + ab + ac + ad + bc + bd + cd\) is not less than 10, engaging with concepts from algebra and inequalities.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses difficulty in transitioning from quadratic to quartic inequalities and seeks assistance in proving a specific inequality.
  • Another participant suggests using the arithmetic and geometric mean inequality as a potential approach to the problem.
  • A later reply clarifies that the initial interpretation of the problem may have been slightly off, hinting at the need for a square root sign.
  • Participants discuss the general form of the arithmetic mean-geometric mean inequality and its simplification when the product is constant.
  • One participant shares a specific manipulation of terms to maintain a constant product, illustrating a technique to approach the inequality.
  • Another participant confirms their solution by applying the AM-GM inequality to the relevant terms and expresses interest in finding a more elegant solution.
  • Further contributions suggest applying the AM-GM inequality directly to all ten terms involved in the expression.
  • Participants reflect on the challenges of switching perspectives when stuck on a problem, emphasizing the commonality of such experiences in mathematical problem-solving.

Areas of Agreement / Disagreement

Participants generally agree on the utility of the arithmetic and geometric mean inequality in addressing the problem, but there is no consensus on a singular approach or the most elegant solution. The discussion remains exploratory with multiple methods proposed.

Contextual Notes

Some participants mention specific manipulations and theorems without fully resolving the mathematical steps involved, leaving certain assumptions and dependencies on definitions unaddressed.

hwmaltby
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Hello!

I normally consider myself fairly decent with algebra, but when it comes to inequalities... Well, I cannot even solve this very simple one, so please help me!

I have been (somewhat) studying inequalities for the past few days, but I cannot find a theorem that deals with changing from quadratics to quartics (by this I mean two variables multiplied together and four variables multiplied together -- not the traditional x^2 or x^4) as is required by this problem:

Show that for positive a, b, c, and c, such that abcd=1, a^2 +b^2+c^2+d^2 + ab+ac+ad+bc+bd+cd is not smaller then 10.

I think it's fairly clear we need to prove:
a^2 +b^2+c^2+d^2 + ab+ac+ad+bc+bd+cd >= 10abcd=10

If you use any theorems, please mention what they are called. If you derive everything from scratch, all the better!

Thank you for any help or hints you can give me!
 
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Oh, there is an general easy trick. Think of the inequality between the arithmetic and geometric mean.
Let me know if you need more hints ;)

PS: What you think is "fairly clear" is close, but not quite what you will need :)
 


Yes, I suppose I forgot a square root sign... That solves my "quartic" problems.
 


Square root sign? Actually there is no need for a modification.
In general [tex]\frac{\sum a_n}{N}\geq \sqrt[N]{ \prod a_n}[/tex]
If the product happens to be a constant, it simplifies a lot.
 


And note that you can use tricks to make the product a constant!
For example
[tex]a^2+\frac{1}{a}=a^2+\frac{1}{2a}+\frac{1}{2a}[/tex]
and there we go again... :)
 


Oh, I got it! Thank you for your help.

I did it by applying AM-GM to a^2, b^2, c^2, and d^2 and again to ab, ac, ad, bc, bd, and cd. I then added them together.

If there is a more elegant way, could you share it?

Once again, thank you for your help!
 


The simplest is to apply
[tex]\sum a_n\geq N\sqrt[N]{\prod a_n}[/tex]
straight to all 10(!) terms ;)
 


Oh, you're right! Wow, I feel extremely stupid now. Well, thank you for all this!
 


Oh, don't worry. It happens to all of us that if you get stuck on the wrong track it's hard to switch to a different idea. Your initial idea was close and tempted you to see it a particular way.
 

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