March 11th, 2017, 12:21 AM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 240 Thanks: 23 Math Focus: Number theory  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?

March 11th, 2017, 04:37 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,172 Thanks: 1285 
No.

March 11th, 2017, 03:59 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 240 Thanks: 23 Math Focus: Number theory 
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. 
March 11th, 2017, 04:35 PM  #4 
Senior Member Joined: Aug 2012 Posts: 1,373 Thanks: 322 
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.

March 11th, 2017, 09:23 PM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 240 Thanks: 23 Math Focus: Number theory 
Such adjusting leaves by far most cubics with the given proportion a constant 2/3 (rationalcountable, not irrationaluncountable). 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. 
March 11th, 2017, 09:31 PM  #6  
Senior Member Joined: Aug 2012 Posts: 1,373 Thanks: 322  Quote:
Can you outline the proof for the cubic?  
March 11th, 2017, 09:40 PM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 240 Thanks: 23 Math Focus: Number theory 
The proof appears as "The Booda Theorem" at http://www.quantumdream.net.

March 12th, 2017, 09:50 AM  #8 
Senior Member Joined: Aug 2012 Posts: 1,373 Thanks: 322 
Did you try it for 4th powers? Looks like a lot of messy algebra but perhaps a computer algebra system would be helpful.

March 12th, 2017, 10:16 AM  #9 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 240 Thanks: 23 Math Focus: Number theory 
Computer programmers have helped me on this forum before, but I don't have the skills needed.

March 12th, 2017, 11:05 AM  #10  
Senior Member Joined: Aug 2012 Posts: 1,373 Thanks: 322  Quote:
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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