Re: pain in neck calculus problem
From: Robert Israel (israel_at_math.ubc.ca)
Date: 06/27/04
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Date: 27 Jun 2004 21:30:45 GMT
In article <iYqDc.116$dc5.74@news01.roc.ny>,
Troubled <mkajumap@hotmail.com> wrote:
>I need to find a 4th degree polynomials with 3 extrema and 2 points of
>inflection, all 5 numbers being integers. I started off with letting y' = x
>(x-p) (x-q) , with p and q integers (so the critical values will be 0,p,and
>q). Then y" = (x-p)(x-q) +x(x-q) + x(x-p). I concluded that y"=0 when [(p+q)
>+/- sqrt( (p+q)^2-3pq)]/3. I know that (p+q)^2-3pq must be a perfect
>square, I^2. I concluded that p= [q +/-sqrt( (2I)^2 - 3q^2)]/2. This is
>where I got stuck. At this point I'd pick values for q and I and see if p is
>an integer and if it was I'd check to see if it satisfied the equation
>above.
With the help of Maple's isolve command, I find that if
p = 2 a b - b^2 and q = a^2 - b^2, (p+q)^2 - 3pq is a square.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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