Need Help with a mathematical modeling question

[shadedude123]

This question deals with a 1-parameter equation:

f_c(x)= (4/x)+(x/2)-c for 0≤c≤ 2.3

a.) Use algebra to find the positive fixed point p(c) of f_c(x) (in terms of c) and identify its exact interval of existence

a.) Ok so for the fixed point I got p(c)= -c+√(c²+8) I'm not sure if this is correct though. I used the quadratic formula to find this after setting the initial equation equal to x and solving in terms of c. Now for the interval of existence I need to take the first derivative of the initial equation which I have as -4x^-2+(1/2). Is this right? And I think I'm supposed to plug in the fixed point i have into this first derivative to get the interval of stability for c by setting an inequality up like: -1<c<1. This would be the interval of existence for the parameter c right? Please correct me if I'm making any mistakes. I can't do this last step because i end up getting ridiculous numbers.

 

[shadedude123]

bump, anyone?

 

[zsviben]

Hi,
I do not know if you still need the help, but i found it interesting to see the answer :)

so from
f(x) = x
or
4/x+x/2-c = x
we have
x^2+2cx-8 = 0
or
x = p(c) = -c+-sqrt{c^2+8}
and since we are only looking for the positive solution we choose
p(c) = -c+sqrt{c^2+8}for the interval we choose the c such that the solution of x=f(x) exists
or in other words that
-1 < f'(x) < 1 , where x = p(c)

Since
f'(x) = -4/(x^2)+1/2
we have

-1 < -4/(2 c^2+8-2c sqrt{c^2+8})+1/2 < 1

from where we obtain
0 < c < 2 sqrt{2/3}

I think this is correct, but if you stil need it I could check and make a better calculation.

 

[Ray Vickson]

This question deals with a 1-parameter equation:

f_c(x)= (4/x)+(x/2)-c for 0≤c≤ 2.3

a.) Use algebra to find the positive fixed point p(c) of f_c(x) (in terms of c) and identify its exact interval of existence

a.) Ok so for the fixed point I got p(c)= -c+√(c²+8) I'm not sure if this is correct though. I used the quadratic formula to find this after setting the initial equation equal to x and solving in terms of c. Now for the interval of existence I need to take the first derivative of the initial equation which I have as -4x^-2+(1/2). Is this right? And I think I'm supposed to plug in the fixed point i have into this first derivative to get the interval of stability for c by setting an inequality up like: -1<c<1. This would be the interval of existence for the parameter c right? Please correct me if I'm making any mistakes. I can't do this last step because i end up getting ridiculous numbers.

I don't understand the question. You have found a solution of x = f(x) in terms of c, and it exists and is positive for all real values of c.

If you don't trust algebra and logic, try instead to use plotting: you can plot f(x) for any given value of c and see right away that the line y = x cuts the graph y = f(x) in just one point x > 0.

RGV