Find two points with opposite x-values?

• Antebellum
In summary, the conversation is about finding the points on the curve y = x^3 / (1 + x^4) with opposite x-values that make the slope of the line joining them greatest. The participants discuss finding the slope of the line and using it to find the maximum slope, with suggestions to take the derivative and find the maxima of a function. Eventually, the problem is solved and the person expresses gratitude for the help.

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!

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

What does it mean to find two points with opposite x-values?

Finding two points with opposite x-values means finding two points on a graph that have the same y-value but different x-values. The x-values will be on opposite sides of the y-axis.

Why is finding two points with opposite x-values important in science?

This concept is important in science because it allows us to identify and analyze relationships between variables. By finding two points with opposite x-values, we can determine if there is a causal relationship between the two variables.

How do I find two points with opposite x-values?

To find two points with opposite x-values, you can plot the points on a graph and look for two points that have the same y-value but different x-values. Alternatively, you can also use algebra to solve for the x-values given the y-value.

What does it mean when two points have opposite x-values but the same y-value?

When two points have opposite x-values but the same y-value, it means that those two points are symmetric to each other across the y-axis. In other words, they are mirror images of each other.

Can two points have opposite x-values and different y-values?

Yes, it is possible for two points to have opposite x-values and different y-values. This means that they are not symmetric to each other across the y-axis, but they still have different x-values and may have a relationship between the two variables being studied.