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March 10th, 2017, 11:21 PM   #1
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Slope ratio related to f'(x) and f"(x)

Is the proportion between the slope of a line connecting a local maximum and minimum, and the nonzero slope of an adjacent inflection point always rational?
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March 11th, 2017, 03:37 AM   #2
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No.
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March 11th, 2017, 02:59 PM   #3
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For a cubic polynomial with local maximum and minimum, the slope of the line connecting them is 2/3 of the slope at the inflection point.

I am trying to generalize this hypothesis of its proof I had once demonstrated.
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March 11th, 2017, 03:35 PM   #4
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I don't see any reason for this to be true. Just start drawing curves. There's no reason for your conclusion unless there are some fairly rigid hypotheses I'd think. You can always adjust the max or min or inflection point to make the ratio irrational.
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March 11th, 2017, 08:23 PM   #5
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Such adjusting leaves by far most cubics with the given proportion a constant 2/3 (rational-countable, not irrational-uncountable).

Why wouldn't this proportion be rational for other powers with a similar rule? Of course this needs to be proved like it was in the cubic case.
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March 11th, 2017, 08:31 PM   #6
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Such adjusting leaves by far most cubics with the given proportion a constant 2/3 (rational-countable, not irrational-uncountable).

Why wouldn't this proportion be rational for other powers with a similar rule? Of course this needs to be proved like it was in the cubic case.
So these are polynomials? That's a pretty strong restriction, gives us something to work with.

Can you outline the proof for the cubic?
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March 11th, 2017, 08:40 PM   #7
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The proof appears as "The Booda Theorem" at http://www.quantumdream.net.
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March 12th, 2017, 09:50 AM   #8
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Did you try it for 4th powers? Looks like a lot of messy algebra but perhaps a computer algebra system would be helpful.
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March 12th, 2017, 10:16 AM   #9
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Computer programmers have helped me on this forum before, but I don't have the skills needed.
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March 12th, 2017, 11:05 AM   #10
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Computer programmers have helped me on this forum before, but I don't have the skills needed.
What do you think might be the form of the conjecture? 4th degree polynomials can have 3 max/minima and 2 inflection points, right? But by your example it looks like there might well be some interesting algebraic relationships among them.

I didn't mean you should write the code yourself. There are computer algebra systems that do all the manipulations for you. https://en.wikipedia.org/wiki/Computer_algebra_system
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