The Jomo Problem

J

JeffM

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Sep 14, 2012
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This has grown out of a student's problem and relates to a question raised by jomo. I do not, or at least not yet, have an answer for it.

Consider the family of functions such that if f(x) belongs to the family

f(x) is a polynomial of even degree with real coefficients;

f(x) has real zeroes at x = - 3, - 1, and 3;

f(0) = 3, and

f(x) has exactly 3 local extrema that are real, a local maximum at - 3, a local minimum in the interval (- 3, - 1). and a local maximum in the interval (- 1, 3)

The question is whether it is mathematically possible for f(x) to have a local maximum in the interval (- 1, 0], and if so, what is the smallest degree of that polynomial?

What has been, I believe, demonstrated so far is that, if such a polynomial exists, it will have to be of degree 6 or higher and will have to be factorable into quadratics of which at least one has no real zeroes.
 
I think the "Jomo problem" is that Jomo keeps answering questions before I can! (And more recently there is the "Dr. Peterson problem".)
 
JeffM said:
This has grown out of a student's problem and relates to a question raised by jomo. I do not, or at least not yet, have an answer for it.

Consider the family of functions such that if f(x) belongs to the family

f(x) is a polynomial of even degree with real coefficients;

f(x) has real zeroes at x = - 3, - 1, and 3;

f(0) = 3,
Click to expand...
So f(x)= a(x+ 3)(x+ 1)(x- 3) and f(0)= a(3)(1)(-3)= -9a= 3 so a= -1/3.
f(x)= -(1/3)(x+ 3)(x+ 1)(x- 3)p(x) for some polynomial p.
Since f is of even degree, the multiple must be of odd degree and any first degree factors must be (x+ 3), (x+ 1), and (x- 3). But there may be irreducible quadratic factors.

and f(x) has exactly 3 local extrema that are real, a local maximum at - 3, a local minimum in the interval (- 3, - 1). and a local maximum in the interval (- 1, 3)

The question is whether it is mathematically possible for f(x) to have a local maximum in the interval (- 1, 0], and if so, what is the smallest degree of that polynomial?

What has been, I believe, demonstrated so far is that, if such a polynomial exists, it will have to be of degree 6 or higher and will have to be factorable into quadratics of which at least one has no real zeroes.
Click to expand...
 
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