Problem 65 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that if \(f\) is continuous on \([0,1]\) and satisfies \(0 \leq f(x) \leq 1\) there, then \(f\) has a fixed point; that is, there is a number \(c\) in \([0,1]\) such that \(f(c)=c\). Hint: Apply the Intermediate Value Theorem to \(g(x)=x-f(x)\).

Step-by-Step Solution

Answer

The function \(f(x)\) has a fixed point \(c\) in \([0,1]\) where \(f(c)=c\).

Step 1: Define the function

We define the function \( g(x) = x - f(x) \). This function will help us find a fixed point for \( f(x) \).

Step 2: Analyze endpoints

Evaluate \( g(x) \) at the endpoints of the interval. At \( x = 0 \), \( g(0) = 0 - f(0) = -f(0) \), so \( g(0) \leq 0 \). At \( x = 1 \), \( g(1) = 1 - f(1) \), hence \( g(1) \geq 0 \).

Step 3: Apply the Intermediate Value Theorem

Since \( g(x) \) is continuous (as a composition of continuous functions \( x \) and \( f(x) \)), and \( g(0) \leq 0 \) while \( g(1) \geq 0 \), the Intermediate Value Theorem implies there exists a \( c \) in \([0,1]\) such that \( g(c) = 0 \).

Step 4: Conclude \( f(c) = c \)

Since \( g(c) = 0 \), it follows from the equation \( g(x) = x - f(x) \) that \( c - f(c) = 0 \) or \( f(c) = c \). Thus, \( c \) is the fixed point of \( f \).

Key Concepts

Continuous functions

Continuous functions play a crucial role when it comes to understanding a variety of calculus concepts. A function is considered continuous on an interval if there are no breaks, jumps, or holes in its graph within that interval. This means that small changes in the input of the function result in small changes in the output. In technical terms, a function \( f(x) \) is continuous at a point \( c \) if the limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \). Continuity on an interval involves being continuous at every point on that interval.

The importance of continuity can be understood beyond the mere absence of interruptions in a function. It lays the groundwork for the application of key theorems in calculus, such as the Intermediate Value Theorem. Additionally, continuous functions allow for intuitive interpretations and predictions of behavior over specific intervals, which is why in this problem, it's essential that \( f \) is continuous over the closed interval \([0, 1]\).

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that illustrates one of the powerful results guaranteed by continuous functions. According to this theorem, if a function \( f \) is continuous on a closed interval \([a, b]\) and \( N \) is a number between \( f(a) \) and \( f(b) \), then there exists at least one number \( c \) in the interval \([a, b]\) such that \( f(c) = N \).

This theorem is crucial for proving the existence of solutions because it assures us that all real numbers within the range of the function can be achieved. In the original exercise, the function \( g(x) = x - f(x) \) is continuous, and we utilize the IVT to show that there is a point \( c \) where \( g(c) = 0 \). This ultimately leads us to find the fixed point, where the function's value matches the input.

Fixed points in calculus

A fixed point of a function occurs when the function's output is equal to its input, denoted mathematically as \( f(c) = c \). These points are valuable in mathematics because they often represent equilibrium states or steady conditions in various mathematical models. In calculus, finding fixed points can provide insights into the behavior and stability of functions and dynamic systems.

In the exercise, the function \( f \) is trapped within the interval \([0, 1]\), with the assurance that a fixed point exists due to the continuous nature of \( f \) and the properties of the Intermediate Value Theorem. By demonstrating that \( g(c) = 0 \), where \( g(x) = x - f(x) \), we see that \( f(c) = c \). Understanding fixed points is vital, as they are not only pivotal in pure mathematics but also applicable in practical areas such as computational algorithms and the study of differential equations.