A Cubic Function and a Straight Line with One-variable Calculus

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Discussion Overview

The discussion revolves around a mathematical problem involving a cubic function and a straight line, specifically focusing on a particular part of the problem related to finding relationships between variables using one-variable calculus. Participants explore various approaches to solving the problem, including algebraic manipulations and coefficient comparisons.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant suggests that the expression x^3 - x - m(x - a) represents the distance from the cubic function to the line, indicating that it is negative for points in the interval (b, c).
  • The same participant attempts to show that c = -2b through algebraic manipulation, leading to the equation b^2(3c - 2b) = c^3, but expresses uncertainty about its usefulness.
  • Another participant proposes a simpler method by suggesting to expand the right-hand side of the given equation and equate coefficients of x^2, implying this could lead to a solution.
  • A third participant expresses surprise at the simplicity of the proposed method and reflects on their own more complicated approach to the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the problem, with multiple methods being discussed and no clear agreement on which is superior.

Contextual Notes

Participants express uncertainty about the effectiveness of their algebraic manipulations and the implications of their findings, indicating potential limitations in their approaches.

seniorhs9
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Hi. I posted this in the Homework question but after 152 views with no right answer, my question looks analytical and rigorous enough to be posted here.

Thank you...

----

I am asking about part iv).

[PLAIN]http://img715.imageshack.us/img715/7977/113ivb.jpg

Attempt at a solution

In the given fact, I think x^3 - x - m(x - a) distance from the cubic function to the line. So for every point in (b, c), this would be negative.

I'm really not sure how to show c = -2b so I just tried to play with some algebra...

At x = b... b^3 - b - m(b - a) = 0

At x = c... c^3 - c - m(c - a) = 0

So they're both equal to 0...


b^3 - b - m(b - a) = c^3 - c - m(c - a)

so b^3 - b - mb = c^3 - c - mc

so by part i) b^3 - b - b(3b^2 - 1) = c^3 - c - c(3b^2 - 1)

so b^3 - b - 3b^3 + b = c^3 - c - 3b^2c + c

so -2b^3 = c^3 - 3b^2c

so b^2(3c - 2b) = c^3

but this doesn't look useful...

Thank you.
 
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FOR PART 4:
You just expand RHS of give eq. in and equate coefficients of x^2 on both sides and you'll get it.
 
Hi omkar13. Thanks so much for your answer.

Now I feel like an idiot...how did you see to expand the RHS of the given eq.?

Somehow I didn't see it...I thought there was something harder going on...
 

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