Proving the Existence of Fixed Points in Monotone Increasing Functions

In summary, the conversation discusses the concept of a fixed point in a function and how it is linked to the constraint on the differential of the function. The group discusses using the Mean Value Theorem and the Intermediate Value Theorem to prove the existence of a fixed point in a monotone increasing function, and how this can be extended to any continuous function. They also discuss using the function g(x) = f(x) - x to prove the existence of a solution to g(x) = 0. Finally, they address a specific step in the proof and clarify its reasoning.
  • #1
zolit
6
0
I'm trying to work out how an existence of a fixed point is linked to the constraint on the differential of that function.

For example, i need to prove f has a fixed point if f'(x)=>2.

I understand that what I have is a monotone increasing function so it is 1-1. All the fixed points are on the line f(x) = x. So conceptually it must be true that these two lines should intersect somewhere, but I can't prove this rigorously.

I have a feeling I should be using the Mean Value Theorem ( f(b) - f(a)) = f'(c)(b-a) but can't get much further than that.
 
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  • #2
You want to prove that there is some c such that f(c) = c. f is continuous on R (I'm guessing). Now you should be able to prove that there are contradictions if f(x) > x for all x, or f(x) < x. Then, by intermediate value theorem, you should be able to finish the proof.
 
  • #3
let g(x) = f(x) - x. then you want to prove that g(x) = 0 has a solution for some x.

But you, know that g'(x) >= 1, so for every pair of successive integers a = n,b = n+1, it follows from MVT that g(b)-g(a) >= 1. Do you see why?

Hence if g(0) = c>0 say, and n < a < n+1, then g(-n-1) < 0. Do you see why?

then since g is differentiable it is also contrinuous, and has both a negative value and a positive value, hence is somewhere zero.

similar arguments work for g(0) = c<0.
 
  • #4
I was reading through that proof, and I did not understand how

"Hence if g(0) = c>0 say, and n < a < n+1, then g(-n-1) < 0" this step came to be?
thanx
 

1. What is a fixed point in a function?

A fixed point in a function is a value where the input and output of the function are equal. In other words, when the input is substituted into the function, the output remains the same.

2. Why is it important for a function to have a fixed point?

A fixed point in a function is important because it helps us find the solution to certain equations or systems of equations. It also allows us to analyze the behavior of a function and make predictions about its behavior.

3. How do you determine if a function has a fixed point?

To determine if a function has a fixed point, we can set the function equal to its input and solve for the variable. If the solution is equal to the input, then the function has a fixed point. Another way is to graph the function and see if it intersects with the line y=x, since the points of intersection represent fixed points.

4. Can a function have more than one fixed point?

Yes, a function can have more than one fixed point. In fact, some functions have an infinite number of fixed points. For example, the function f(x)=x^2 has fixed points of 0 and 1, but also has an infinite number of fixed points between 0 and 1.

5. What is the significance of a function having no fixed points?

A function with no fixed points means that there is no solution to the equation f(x)=x. This can indicate that the function has no real roots or that the roots are complex numbers. It can also mean that the function is constantly increasing or decreasing and never crosses the line y=x.

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