Find two points with opposite x-values?

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Homework Help Overview

The problem involves finding two points on the curve defined by the function y = x^3 / (1 + x^4) that have opposite x-values, specifically x and -x, and determining the points that maximize the slope of the line connecting them.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster expresses confusion about how to start solving the problem and considers taking the derivative to find critical points. They also question whether graphing the function is necessary.
  • Participants suggest writing down the equation of the line joining the points and finding its slope, which leads to further exploration of the relationship between the slope and the function.
  • There is a discussion about the need to find the maximum slope and how to approach that mathematically.

Discussion Status

Participants are actively engaging with the problem, with some offering guidance on how to derive the slope and suggesting methods to find maxima. The original poster has made progress but is still seeking clarity on the next steps.

Contextual Notes

The original poster indicates uncertainty about the necessary equations and methods, highlighting a lack of clear direction in their initial approach to the problem.

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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 [itex]f(x) = x^3 / (1 + x^4)[/itex].
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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