**Mathematical Problem Statement:**Prove the inequality:\[ d(x, y) \leq d(x, r) + d(r, y) \]where \( d(x, y) = (x - y)^4 \).**Explanation:**The given expression is considering a metric space condition. The function \( d(x, y) = (x - y)^4 \) is defined as the distance between two points \( x \) and \( y \).The goal is to verify if this function satisfies the triangle inequality, which is a fundamental property of metric spaces. In a typical metric space, for any points \( x \), \( y \), and \( r \), the triangle inequality must hold:\[ d(x, y) \leq d(x, r) + d(r, y) \]For this specific function, we are required to check this condition for the expression \( (x - y)^4 \).**Approach:**1. **Substitute:** Substitute \( d(x, y) = (x-y)^4 \), \( d(x, r) = (x-r)^4 \), and \( d(r, y) = (r-y)^4 \) into the inequality. 2. **Simplify and Analyze:** Compare \( (x-y)^4 \) with \( (x-r)^4 + (r-y)^4 \) to determine if the inequality holds for all real numbers \( x \), \( y \), and \( r \).3. **Discussion:** Discuss whether \( (x-y)^4 \) inherently satisfies the properties required for a metric, particularly focusing on non-negativity, symmetry, and the triangle inequality. Insights or conclusions will reveal whether this expression is a valid metric.This process will help determine if the given function is a valid metric by testing it against the axioms of metric spaces.

Answered: prove d(x, y) ≤ d (x, r) + d(r, y)…

prove d(x, y) ≤ d (x, r) + d(r, y) where d(x, y) = (x-y)4

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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