A Lesson in Squares

**Akzis** · 06-29-2012, 08:11 PM

Let a, b, and c be real numbers such that a^2 + c^2 <= 4b. Prove that for all x in R, x^4 + ax^3 + bx^2 + cx + 1 >= 0.

Hint: think absolute values...

**The Dan Keizer** · 06-29-2012, 08:59 PM

Is there a reason I would do that?

**Monte314** · 07-01-2012, 07:55 AM

OK, I got it. My proof does not use the method suggested in the OP, but relies on elementary techniques and the Intermediate Value Theorem (studied in Calculus I).

I'll post it if no one else posts a proof.

22 hours later...
*crickets chirping*

**Monte314** · 07-02-2012, 09:46 AM

Here is my solution. It is neither insightful nor beautiful; I was going for the "quick kill".

( Show Spoiler )

My work is not totally uncool, however. I wrote a little computer program to actually step through a range of values of a, b, and c, plotting the resulting Q(x) to see whether it dipped below the x-axis.

Here is a movie I made of my test that shows 62,500 instances being plotted on the same set of axes in just a few seconds. The green horizontal line is the x-axis... the result survives an empirical test!

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06-29-2012, 08:11 PM	#1
Akzis Member [36%] MBTI: INTP Join Date: Apr 2010 Posts: 1,450	Let a, b, and c be real numbers such that a^2 + c^2 <= 4b. Prove that for all x in R, x^4 + ax^3 + bx^2 + cx + 1 >= 0. Hint: think absolute values...
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06-29-2012, 08:59 PM	#2
The Dan Keizer Core Member [110%] MBTI: INTJ Join Date: Jun 2012 Posts: 4,402	Is there a reason I would do that?
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07-01-2012, 07:55 AM	#3
Monte314 Core Member [407%] MBTI: INTJ Join Date: Apr 2008 Posts: 16,303	OK, I got it. My proof does not use the method suggested in the OP, but relies on elementary techniques and the Intermediate Value Theorem (studied in Calculus I). I'll post it if no one else posts a proof. 22 hours later... crickets chirping Last edited by Monte314; 07-02-2012 at 06:40 AM.
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07-02-2012, 09:46 AM	#4
Monte314 Core Member [407%] MBTI: INTJ Join Date: Apr 2008 Posts: 16,303	Here is my solution. It is neither insightful nor beautiful; I was going for the "quick kill". ( Show Spoiler ) To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. My work is not totally uncool, however. I wrote a little computer program to actually step through a range of values of a, b, and c, plotting the resulting Q(x) to see whether it dipped below the x-axis. Here is a movie I made of my test that shows 62,500 instances being plotted on the same set of axes in just a few seconds. The green horizontal line is the x-axis... the result survives an empirical test! To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
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