Let f be a continuous function on the closed interval [0,1] with range also contained in [0,1].Prove that f must have a fixed point; that is, show f(x)=x for at least one value of x belong to [0,1].

Solution Preview

Proof:
We consider the function g(x)=f(x)-x
From the condition, the range of f(x) ...

Solution Summary

The solution is comprised of a proof regarding the existence of a fixed point.

(See attached file for full problem description with proper equations)
---
3. Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixedpoint on this interval. Does this conflict Theorem 6.4? Explain.
Note: We are using the book Methods of Real Analysis by Richard R. Goldberg.
This

A number (a) is called a fixedpoint of a function (f) if f(a)=a. Prove that, if f'(x) does NOT equal 1 for all real numbers (x), then f has at most one fixedpoint.

Assume that (X, d) is a compact metric space, and let f: X -> X be a function such that the inequality d(f(x), f(y)) < d(x, y) holds for all distinct elements x, y in X. Show that f has a unique fixedpoint.
See attached file for full problem description.

The real root of x^3 - x - 1 is about 1.3.
a) Construct a fixedpoint iteration formula for finding the root and prove that the formula will work.
b) Construct another fixedpoint iteration formula and prove that it will not work.

I need help on these two problems please
1. Does the existence and value of the limit of a function f(x) as x approaches x_o ever depend on what happens at x = x_o? Explain and give examples.
2. What does it mean for a function to be continuous? Give examples to illustrate the fact that a function that is not continuous on

FixedPoint iteration method.
Use a fixed-point iteration method to find an approximation to that is accurate within 10-4
See attached file for full problem description.

Some firms have a lot of fixed costs and few variable costs, while other firms are configured the other way around. What affect do you think the existence of a high proportion of fixed costs has on the desirability of using ABC methods?

1. The equation x - 3 ln x = 2 has exactly two solutions A and B, with 0 < A < B.
(You do not have to show this.)
(a) Show that A is in [0.5,0.7], and B is in [8.3,8.5].
(b) Consider the following fixed-point iteration for finding a solution of the given
equation: xn+1 = e(1/3(Xn-2)):
Show that if X0 = B+ E, where E is