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Traditional x^2 or x^4

  1. Apr 23, 2010 #1
    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!
     
  2. jcsd
  3. Apr 24, 2010 #2
    Re: Inequalities

    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 :)
     
  4. Apr 24, 2010 #3
    Re: Inequalities

    Yes, I suppose I forgot a square root sign... That solves my "quartic" problems.
     
  5. Apr 24, 2010 #4
    Re: Inequalities

    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.
     
  6. Apr 24, 2010 #5
    Re: Inequalities

    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... :)
     
  7. Apr 24, 2010 #6
    Re: Inequalities

    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!
     
  8. Apr 24, 2010 #7
    Re: Inequalities

    The simplest is to apply
    [tex]\sum a_n\geq N\sqrt[N]{\prod a_n}[/tex]
    straight to all 10(!) terms ;)
     
  9. Apr 24, 2010 #8
    Re: Inequalities

    Oh, you're right! Wow, I feel extremely stupid now. Well, thank you for all this!
     
  10. Apr 25, 2010 #9
    Re: Inequalities

    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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