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Proof with Inequalities

Date: 10/10/1999 at 21:28:14
From: James
Subject: Inequality Proof and Problem Solving

Dear Dr. Math,

I wish to query on two questions that I have received. I am not quite 
sure how to solve them and your assistance is appreciated.

Question One

Two real numbers a and b satisfy the equation ab = 1.

1) Prove that a^6 + 4*b^6 >= 4
2) Find out whether the inequality a^6 + 4*b^6 > 4 holds for all a and 
   b such that ab = 1.

I have been shown a proof 'similar' to this one in about 20 minutes, 
which makes it very hard for me to understand fully. The example given 
was the proof that Arithemetic Mean > Geometric Mean.

It involved this:

     (x+y)/2 >= sqrt(xy)


     (x+y) >= 2 * sqrt(xy)

And X was let to be a^2
    Y was let to be b^2

so a^2 + b^2        >= 2 * sqrt(a^2 * b^2)
   a^2 + b^2        >= 2ab
   a^2 - 2ab + b^2  >= 0


Assuming that you have seen this before, I will not go on, but I'm not 
sure how to apply this example to answer my question. I have tried to 
use this example but, I'm not sure how they relate. I have considered 
letting a^6 = x^2 and b^6 = y^2 to try and tackle it similar to the 
Mean Proof, but I am unsure of how to do it. If you can start me off 
and lead me into it, I will be grateful.

Question Two

At each vertex of a polygon, a number is written. No two numbers are 
the same, and each is the product of the numbers at the two vertices 
next to it. How many sides does the polygon have?

I'm not sure what the numbers could be: small/large, in sequence/
not-in-sequence, etc.

Your help is appreciated.

Thank you,

Date: 10/11/1999 at 11:53:10
From: Doctor Rick
Subject: Re: Inequality Proof and Problem Solving

Hi, James. 

You have done an admirable job of showing us how you have tried to 
solve the problems.

Question 1.

Your thoughts about the similarity of this problem to the proof of the 
theorem are good; recalling similar problems is a helpful 
problem-solving method. Note that the proof as you present it is 
backward, assuming the conclusion and finding a way to derive the 
premises of the theorem. This is often a useful way to find a proof, 
much like tracing a maze backward (it is often easier to follow that 
way). When you've found the way, you must retrace the path in the 
correct direction.

The idea behind the proof (in the forward direction) is that the 
square of a real number is always positive, so we can derive an 
inequality involving the terms of the expansion of a square. Thus,

           (a - b)^2 >= 0

     a^2 - 2ab + b^2 >= 0

           a^2 + b^2 >= 2ab

Now let's look at your problem. Look at what you are given, and what 
you need to prove:

     Given: ab = 1
     To prove: a^6 + 4*b^6 >= 4

We would like to construct the expansion of a square, so try to match 
the expression on the left to

     (x + y)^2 = x^2 + 2xy + y^2

You'll get something close to what you tried. Now you have something 
of the form

     (expression in a and b)^2 >= 0

There is your starting point. Follow the procedure of the proof as I 
showed above.

Question 2. How many sides does the polygon have?

This is interesting! Here is what I would do. Imagine you have some 
large polygon. Label two neighboring vertices a and b (these represent 
two DIFFERENT numbers). Find an expression for the third vertex in 
terms of a and b. Continue around the polygon, and see what happens.

I hope these hints help you.

- Doctor Rick, The Math Forum   
Associated Topics:
High School Basic Algebra

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