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February 15th, 2019, 06:36 AM   #11
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 Originally Posted by SDK Everything up to here is fine but this reasoning is completely invalid. There are countless counterexamples and the subject of integer linear programming exists because you can't do optimization on the integers by doing optimization on the reals and then looking "nearby" for an integer solution.
This is defeating my intuition.

$\text {IF } f(x) \text { is continuous on } [u, v], \text { differentiable on } (u, v),$

$u \le x < w \implies f'(x) > 0, \text { and } w < x \le v \implies f'(x) < 0,$

then you say that it may be false that f(x) is maximized on the integers at either the floor or else the ceiling of w. That is essentially the situation on this problem except we have an open interval. I do not see that your counter-example is relevant to this problem.

I can see that things become more complex if there are multiple points in an interval where the function's derivative is zero, but, for a differentiable function on an interval that includes at least one integer, how can the maximum on the integers not be near an endpoint or a critical point?

I don't understand "completely invalid." Perhaps "over simplified" is intended. Or, perhaps, my intuition is playing me completely false.

Last edited by JeffM1; February 15th, 2019 at 06:38 AM. February 15th, 2019, 08:00 AM #12 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond Please remain on topic. February 15th, 2019, 01:30 PM   #13
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 Originally Posted by SDK I'm aware that in this case it happens to give the correct answer. My comment was intended to point out the flaw in logic. If a student turned in something like that with the "right" answer and that explanation it would be graded pretty harshly. Also, it has nothing to do with being a polynomial. One proof that it works in this example easily follows from the AM/GM inequality. So more generally it will work for monomials. I can't see any structure that is more general which would make this still work. It is definitely false for polynomials in general. Here is a simple counterexample. $p(x) = -\frac{118}{75}x^4 + \frac{758}{75}x^3 - \frac{3121}{150}x^2 + \frac{2141}{150}x$ is a polynomial with rational coefficients even. Its easy to check that if you maximize this on $[0,4]$ the maximum is at $x = 2.5$. But $p(2) = p(3) = 1$ are not the maximum over the integers since $p(1) = 2$. This example wasn't particularly hard to generate.
The polynomial you describe has more than one relative maximum in the interval. To get the answer, all of these should be checked. February 15th, 2019, 02:20 PM #14 Senior Member   Joined: Dec 2015 From: somewhere Posts: 634 Thanks: 91 For the third image : Calculate area $\displaystyle S_1$ of curve y1 bounded by x-axis . Calculate the area $\displaystyle S_2$ of line y2 bounded by x-axis and y1 . The enclosed area between y1 and y2 is $\displaystyle S_1 -S_2$ . Thanks from Gersrus71 Last edited by idontknow; February 15th, 2019 at 02:27 PM. February 16th, 2019, 05:37 AM #15 Senior Member   Joined: Dec 2015 From: somewhere Posts: 634 Thanks: 91 You must find the intersection points of y1 and y2 . Just equal them y1=y2 and solve for x. February 16th, 2019, 06:15 AM   #16
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 Originally Posted by idontknow You must find the intersection points of y1 and y2 . Just equal them y1=y2 and solve for x.
Cheers mate but I’m totally lost
I think I’m just gonna give up on Maths February 16th, 2019, 09:00 AM #17 Senior Member   Joined: Dec 2015 From: somewhere Posts: 634 Thanks: 91 Equation y1=y2 has solutions -4 and 3 . $\displaystyle S=\int_{-4}^{3}( y_1 -y_ 2 )dx$ . Thanks from Gersrus71 Tags assignment Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post lilshrekboye29 Calculus 2 February 11th, 2019 01:01 PM mayak201 Physics 5 April 8th, 2018 01:33 PM illidan5 Number Theory 5 March 24th, 2016 04:31 AM LMW Algebra 1 November 20th, 2012 02:46 PM ron Calculus 1 October 8th, 2011 09:36 AM

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