| > Determine whether or not there is a cubic polynomial with three
> distinct real zeros a1, a2, a3 and two critical numbers c1, c2 (a1 < c1
> < a2 < c2 < a3) such that the segment from a1 to a2 is divided
> harmonically by c1 and c2. [If the "cross-ratio"
> (c1-a1)(c2-a2)/(c1-a1)(c2-a1) of a1, a2, c1, and c2 is -1, then c1 and
> c2 are said to divide a1 and a2 harmonically.]
I believe there is a typo, the cross ratio should really be
(c1-a1)(c2-a2)/(c1-a2)(c2-a1).
The condition on the cross ratio is
(c1-a1)(c2-a2) + (c1-a2)(c2-a1) = 0
Let c1 = x0-dx and c2 = x0+dx
(x0-a1-dx)(x0-a2+dx) + (x0-a2-dx)(x0-a1+dx) =
2*(x0-a1)(x0-a2) - 2*dx^2 = 0 [1]
Now, the cubic in terms of its roots is
y = x^3 - (a1+a2+a3)*x^2 + (a1*a2 + a1*a3 + a2*a3) - a1*a2*a3
y' = 3*x^2 - 2*(a1+a2*a3)*x + (a1*a2 + a1*a3 + a2*a3) [2]
The roots of y' are the c1 and c2, or x0-dx and x0+dx
(where x0 happens to be the inflection point of the cubic)
y' = 3*x^2 - 6*x0*x + 3*(x0^2 - dx^2) [3]
Comparing coefficients in [2] and [3]
3*x0 = a1+a2+a3
3*(x0^2 - dx^2) = a1*a2 + a1*a3 + a2*a3
These can be subtituted in [1]
(x0-a1)(x0-a2) - dx^2 = x0^2 - dx^2 - (a1+a2)*x0 + a1*a2 = 0
3*(x0^2 - dx^2) - 3*(a1+a2)*x0 + 3*a1*a2 = 0
a1*a2 + a1*a3 + a2*a3 - (a1+a2)*(a1+a2+a3) + 3*a1*a2 = 0
-a1^2 + 2*a1*a2 - a2^2 = 0
(a1-a2)^2 = 0
But this contradicts that a1 and a2 are distinct.
- Jim
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