Solve Inequalities with Traditional x^2 or x^4

[hwmaltby]

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!

 

[Gerenuk]

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 :)

 

[hwmaltby]

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

 

[Gerenuk]

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

 

[Gerenuk]

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

 

[hwmaltby]

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!

 

[Gerenuk]

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

 

[hwmaltby]

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

 

[Gerenuk]

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.