How parameter affects the roots of the equation.

[tysonk]

I have trouble figuring out this problem. If someone can help me out that would be appreciated.
Suppose we have x^3 -x + C
let r denote the largest root of the equation.
What's a mathematical way of relating r and C?
(quantitative).
Also what's dr/dc

Thanks in advance.

 

[Mark44]

I have trouble figuring out this problem. If someone can help me out that would be appreciated.
Suppose we have x^3 -x + C
let r denote the largest root of the equation.

What you have above is not an equation.
Did you mean x^3 - x + C = 0? That's an equation.

What's a mathematical way of relating r and C?

Just as a guess, you would have to find the solutions to the cubic equation x^3 - x + C = 0 (assuming that's the equation here).

Depending on the value of C, there will be:

• 3 real solutions
• 2 real solutions, with one that is repeated
• 1 real solution, and two complex solutions

When you say that r is the largest solution, do you mean largest real solution?

(quantitative).
Also what's dr/dc

Thanks in advance.

 

[tysonk]

Thanks for the reply.
Yes, largest real root. The equation you have is correct. So how would I got about finding r (or expressing it). And then also finding the derivative. I suspect that it has something to do with linear approximations or Newton's Method but am not sure how to go about doing it. Any help is much appreciated.

 

[Mark44]

I think you need to find the roots of the equation before you can decide which of them is the largest. Here's a link to a wiki page on cubic equations: http://en.wikipedia.org/wiki/Cubic_function.

 

[tysonk]

Thanks for your reply.
Hmm looking back at the question. It just says "give some quantitative information about r."
"Calculate dr/dc"

For dr/dc I did,

y=x^3 -x + c =0

r^3 -r +c = 0

finding the derivative of that i got
dr/dc = 1/ (3r^2 -1)
I used the fact that c is not a constant yet it is a function of r... I'm not sure if I'm right.

 

[Mark44]

It should be dr/dc = -1/(3r^2 - 1)