Find two points with opposite x-values?

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SUMMARY

The discussion centers on finding two points on the curve defined by the function f(x) = x^3 / (1 + x^4) that have opposite x-values, specifically x and -x, while maximizing the slope of the line connecting them. The slope of the line was determined to be x^2 / (1 + x^4). Participants suggested taking the derivative to find critical points for maximizing the slope, leading to a successful resolution of the problem. The collaborative effort highlighted the importance of deriving the slope equation and understanding how to find maxima in calculus.

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  • Understanding of calculus concepts, specifically derivatives and critical points
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  • Knowledge of function behavior and graphing techniques
  • Basic algebra skills for manipulating equations
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  • Study the concept of slope between two points on a curve
  • Explore optimization techniques in calculus
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Students studying calculus, particularly those focusing on optimization problems and slope calculations, as well as educators looking for collaborative problem-solving techniques in mathematics.

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Homework Statement



Two points on the curve y = x^3 / (1 + x^4) have opposite x-values, x and -x. Find the points making the slope of the line joining them greatest.

Homework Equations



Not certain which equations I'd need here, because I don't know how to begin solving this...

The Attempt at a Solution



I got nothin'. :( I'm really puzzled by this one and don't even know how to start to solve it.

I thought of taking the derivative and finding the critical points (maxima/minima), then seeing if they had opposite x-values, but that certainly wouldn't work, it doesn't even make sense.

Do I need to graph the function and figure it out from that? Somehow I get the feeling that I can solve this without graphing it, though.

I'm definitely not asking for a solution because I want to understand this problem... just point me in the right direction to get started? I'd appreciate it very much! Thank you!
 
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You are asked about the line joining the points at (x, f(x)) and (-x, f(-x)) on the graph of the function f(x) = x^3 / (1 + x^4).
So how about you try to write down the equation of that line?
In fact, let's start by finding the slope.
 
Thanks CompuChip! I did what you suggested and ended up getting x^2 / (1 + x^4) as the slope of the line, and y = (x^2 / (1 + x^4))x + b as the equation of the line.

Still not seeing what I need to do next, though. :( I'm not understanding how to find b, or how I get x and -x out of this...
 
The question asked you to find the maximum slope.

RGV
 
Do I need to start by taking the derivative? I found the slope of the line joining x and -x, and now I'm not sure what to do with it.
 
You know a function for the slope. Maybe you know a way to find the maxima of a function?
 
Sorry for replying so late to you all - I ignored my homework over Thanksgiving break, lol - but just wanted to say that I finally figured out how to solve this problem. YAY. Thank you so much to everyone who nudged me in the right direction. :D
 

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