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Help on polynomial functions, please!

Hello everyone, I hope you can help me out. I'm typing this in a rush so please forgive me for not using LaTeX. Here is the problem:

Given the function f(x)=-0.00128x^3+0.144x^2-4.4x+60, find the point on f(x) whose coordinates have the greatest possible product.


If it's gonna be any help, here are the coefficients in fraction form (in the same order): -4/3125, 18/125, -22/5, and 60.

The graph does not pass through QIV, so I'm assuming the required point is in QI because points in QII and QIII have coordinates with opposite signs and points on the axes have at least one zero coordinate. That's easy enough, but I'm stumped beyond that point. Help in any form would be greatly appreciated. Thank you very much!

PS This is not a school assignment. I'm working as a math editor for a textbook publishing company. This problem came up, and I have to crack it to make sure that the problem is solvable. Too bad I don't see how it can be done. I really hope you can help me, but more than that, I hope this post doesn't get junked for violating Brilliant guidelines. Thanks again!

Note by Francis Gerard Magtibay
10 months, 1 week ago

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What have you tried? Where did you get stuck?

  1. How would you express the product of the coordinates?
  2. How do you find the maximum of this product?
Calvin Lin Staff · 10 months, 1 week ago

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@Calvin Lin First I tried multiplying f(x) by x to obtain another function that yields the product of the coordinates (let's call it g(x)), then I differentiated that function. So now I have g'(x). That's all I've done so far, but I'm not sure if it's correct. I'd continue by using derivatives to find maxima and minima, but then the problem would be off-topic and we'll have to transfer it to a more advanced section. Anyway, this is an interesting refresher because I haven't gone beyond algebra in a while. Thanks Sir Calvin! Francis Gerard Magtibay · 10 months, 1 week ago

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@Francis Gerard Magtibay That's certainly the first approach that I would have taken, given that it is a "textbook question". However, it doesn't have a nice simple solution, and I would resort to using a calculator (not sure if that is allowed).

In what section did they place this under? Is there the possibility of a typo? Calvin Lin Staff · 10 months ago

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@Calvin Lin Actually they put it in a test about polynomial functions, so it's really way too advanced. The other questions in the test are fine because calculators are allowed, but somehow a calculus problem found its way in. I'll recommend to have that last question removed.

Here is an overview of the given information. (I skipped the backstory, but the questions are copied verbatim.)

The following ordered pairs represent the number of kg of mangoes sold as a function of the price per kg: (50, 40), (0, 60), (20, 20), (75, 0).

  1. Find the cubic function f(x) that satisfies the given information.
  2. Use f(x) to predict the number of kg sold at a price of 40.
  3. At what price/s can the store owner expect to sell 50 kg?
  4. Find the zeros of f(x).
  5. Use f(x) to determine the price that will yield the greatest sales. (Hint: sales = price*number of kg sold)

EDIT: I'm done solving. Thanks for the hint, I'm getting really rusty. :) Francis Gerard Magtibay · 10 months ago

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