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Given that |f'(x)| < 1 for all real numbers x, show that If(x1) – f(x2)| < \x1 – x2| for all real numbers x1 and x2.

Given that |f'(x)| < 1 for all real numbers x, show that If(x1) – f(x2)| < \x1 – x2| for all real numbers x1 and x2.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Given that \(|f'(x)| \leq 1\) for all real numbers \(x\), show that \(|f(x_1) - f(x_2)| \leq |x_1 - x_2|\) for all real numbers \(x_1\) and \(x_2\). **Explanation:** This problem involves proving a property of a function \(f(x)\) given a condition on its derivative \(f'(x)\). The condition states that the absolute value of the derivative is always less than or equal to 1 for all real numbers \(x\). The goal is to demonstrate that the absolute difference in the function's values \(|f(x_1) - f(x_2)|\) is bounded by the absolute difference in the input values \(|x_1 - x_2|\). Essentially, this establishes that the function does not change too rapidly; in other words, it is a Lipschitz condition with a Lipschitz constant of 1.
Transcribed Image Text:**Problem Statement:** Given that \(|f'(x)| \leq 1\) for all real numbers \(x\), show that \(|f(x_1) - f(x_2)| \leq |x_1 - x_2|\) for all real numbers \(x_1\) and \(x_2\). **Explanation:** This problem involves proving a property of a function \(f(x)\) given a condition on its derivative \(f'(x)\). The condition states that the absolute value of the derivative is always less than or equal to 1 for all real numbers \(x\). The goal is to demonstrate that the absolute difference in the function's values \(|f(x_1) - f(x_2)|\) is bounded by the absolute difference in the input values \(|x_1 - x_2|\). Essentially, this establishes that the function does not change too rapidly; in other words, it is a Lipschitz condition with a Lipschitz constant of 1.
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