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February 15th, 2019, 06:36 AM   #11
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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.
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February 15th, 2019, 08:00 AM   #12
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Please remain on topic.
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February 15th, 2019, 01:30 PM   #13
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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.
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February 15th, 2019, 02:20 PM   #14
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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 $\displaystyle S_1 -S_2$ .
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Last edited by idontknow; February 15th, 2019 at 02:27 PM.
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February 16th, 2019, 05:37 AM   #15
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You must find the intersection points of y1 and y2 .
Just equal them y1=y2 and solve for x.
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February 16th, 2019, 06:15 AM   #16
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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
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February 16th, 2019, 09:00 AM   #17
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Equation y1=y2 has solutions -4 and 3 .
$\displaystyle S=\int_{-4}^{3}( y_1 -y_ 2 )dx$ .
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