Finding fixed points non-algebraically

[razmtaz]

Homework Statement

let f = \mue^x
let 0 < \mu < 1/e

Show that f has two fixed points q and p with q < p

Homework Equations

a fixed point p is a point such that f(p) = p

The Attempt at a Solution

solving f(x) = x:
f(x) - x = 0
\mue^x - x = 0

Now I want to take logarithms but ln(0) is undefined.

This is the 'normal', algebraic way of solving for fixed points. Is there another way to solve for fixed points? I tried looking at the derivatives based on a suggestion but f'(x) = f(x) which doesn't tell me very much, and I also tried iterating the function, but it just seems to get messy (ie f(f(x)) = \mue^\muex )

Any suggestions for finding fixed points would be helpful. I would prefer not to use the Newton-raphson method to find a root of g(x) = f(x) - x, so if there are any other strategies please let me know

 

[morphism]

Let g(x)=f(x)-x. A fixed point of f is a zero of g. Now try using the intermediate value theorem.

 

[LCKurtz]

If you solve $\mu e^x = x$ for $\mu$ you get $\mu = xe^{-x}$. What are the min and max values of $xe^{-x}$ for $x\ge 0$? And if you can conclude there is one fixed point and the slopes aren't equal there, the graphs of $x$ and $xe^{-x}$ must cross. Then maybe you can get the other crossing by a concavity argument. Your post doesn't say you must calculate the crossings, just show they exist.

 

[razmtaz]

morphism and LCKurtz thanks for the replies.

Gonna try IVT in a few minutes then ill update, but for LCKurtz suggestion:

Im not really sure what you mean. I found the max of x*e^-x to be 1/e at x = 1 and the min to be 0 at x = 0 (which is what we would expect, given the statement 0<mu<1/e). I am not sure how this ties in with:

And if you can conclude there is one fixed point and the slopes aren't equal there, the graphs of $x$ and $xe^{-x}$ must cross. Then maybe you can get the other crossing by a concavity argument.

the slopes of x and x*e^-x? So if I can find one fixed point of mu*e^x then I can use the fact that the slopes of mu = x*e^-x and x are not equal to conclude that the graphs will cross again and therefore there will be another fixed point? I am having trouble understanding because mu is the parameter and I thought we wanted to look at the original function mu*e^x for varying mu, not solving for mu explicitly, or does it not make a difference?

Thanks for the help

 

[razmtaz]

Okay, I determined a solution using IVT, thanks for the tip morphism. There is another part of the question that asks which point is attracting and which is repelling, and to do this part I want to use a theorem that says if the value of the derivative at a fixed point is <1 then the point is attracting; if it is greater than 1 it is repelling. To do this I need to find the points explicitly.

How would I go about doing that as the IVT doesn't give me anything more than the existence of points, and any other method I can think of to do this would involve Newton method. Should I just use Newton method in this case?

edit: Okay I am stupid, just realized Newton method will approximate the answers for me, but I still need to find them explicitly if I want to use the theorem i quoted. Any ideas on how to find these fixed points? or better yet any different method for determining which point is attracting and repelling without explicitly finding the points? I can use a graphical argument but would prefer to have a mathematical one if possible

 

[Dick]

Okay, I determined a solution using IVT, thanks for the tip morphism. There is another part of the question that asks which point is attracting and which is repelling, and to do this part I want to use a theorem that says if the value of the derivative at a fixed point is <1 then the point is attracting; if it is greater than 1 it is repelling. To do this I need to find the points explicitly.

How would I go about doing that as the IVT doesn't give me anything more than the existence of points, and any other method I can think of to do this would involve Newton method. Should I just use Newton method in this case?

edit: Okay I am stupid, just realized Newton method will approximate the answers for me, but I still need to find them explicitly if I want to use the theorem i quoted. Any ideas on how to find these fixed points? or better yet any different method for determining which point is attracting and repelling without explicitly finding the points? I can use a graphical argument but would prefer to have a mathematical one if possible

Nah. I would look at it this way. The fixed points are where g(x)=\mu e^x - x=0. g(x) has a negative minimum at x_0 = log(\frac{1}{\mu}) and approaches \mu as x->0 and infinity as x->infinity. Sketch a graph. So there are two zeros. One is at an x value less than x_0 and one is greater. What's f&#039;(x_0)? What can you say about the derivatives at the two zeros?

 

[razmtaz]

Thanks for the reply Dick.

So if I am understanding you we have a minimum between the two zeroes, at ln(1/mu), and the value of the derivative of mu*e^x at this point is 1; Since we have two values on either side of this point which are higher than it, we can say the following about the derivative:

For the fixed point to the left of the minimum the derivative will have negative slope (since the graph of f is higher/larger at this point). Similarly for the zero to the right it will have positive slope. And there is the answer without explicitly finding the points.

I think that's it, right? Thanks a ton!

 

[Dick]

Thanks for the reply Dick.

So if I am understanding you we have a minimum between the two zeroes, at ln(1/mu), and the value of the derivative of mu*e^x at this point is 1; Since we have two values on either side of this point which are higher than it, we can say the following about the derivative:

For the fixed point to the left of the minimum the derivative will have negative slope (since the graph of f is higher/larger at this point). Similarly for the zero to the right it will have positive slope. And there is the answer without explicitly finding the points.

I think that's it, right? Thanks a ton!

No. That's not quite it. The slope is f&#039;(x) = \mu e^x. That's an increasing function and always positive since \mu &gt; 0. Please rethink that and try again.

 

[razmtaz]

Oops, guess I was thinking about g(x) = mu*e^x - x.

I guess I want to say that for f(x) = mu*e^x there is some fixed point a < x_0 (x_0 is the minimum of g(x)) and some fixed point b > x_0

Since the function is positive and increasing, and at the point x_0 its derivative is equal to 1, then for all input values less than x_0 (including a) its derivative will be less than one, so a will be an attracting fixed point.

Similarly, its increasing, so for values greater than x_0 (including b) it will have slope greater than one -> repelling

 

[Dick]

Oops, guess I was thinking about g(x) = mu*e^x - x.

I guess I want to say that for f(x) = mu*e^x there is some fixed point a < x_0 (x_0 is the minimum of g(x)) and some fixed point b > x_0

Since the function is positive and increasing, and at the point x_0 its derivative is equal to 1, then for all input values less than x_0 (including a) its derivative will be less than one, so a will be an attracting fixed point.

Similarly, its increasing, so for values greater than x_0 (including b) it will have slope greater than one -> repelling

That's more like it.

 

[razmtaz]

Awesome, thanks!