Proof with InequalitiesDate: 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) so, (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 etc. 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, James 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 http://mathforum.org/dr.math/ |
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