**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

Similar questions

(a) f(2, 1, 2) (b) f(-3, 2, 1) (c) f(0, 0, 0) (e) f(t, 1², –t) (d) f(a, a, a) (f) f(a +b, a – b, b). 10. Let f(x, y, z) = zxy + x. Find (a) f(x+ y, x – y, x²) 11. Find F(f(x), g(y), h(z)) if F(x, y, z) = ye*yz, f (x) = x², 8(y) = y + 1, and h(z) = z². 12. Find g(u(x, y, z), v(x, y, z), w(x, y, z)) if 8(x, y, z) = z sin xy,u(x, y, z) = x²z°,v(x, y, z) and w(x, y, z) = xy/z. (b) f(xy, y/x, xz). %3D 2-3 %3D = TXyZ,
w = f (u, v) u = x²y and v = 3x + 2y Let I| se ду Find (1) (2) (3) дхду dy2
The critical points of the function f (x, y) = 1 – x² + 6x – y + 8y2 are | O a) (3,0) and (–3,0) b) None of these O c) (3,0), (3, 2) and (3, –2) O d) (-3, 0), (–3, 2) and (-3, –2)
Let w = f(u), where u = x g(y), show that wxx - f" (x g(y)).g²(y).
If f(x, y) = Select one: O True O False 1 x² + 2xy + y³ 2x + 2y 3y² + 2x (x² + 2xy + y³)²¹ (x² + 2xy + y³)² then Vƒ(x, y) = −(-
Let F: x?+y',x=v - u², and y= df du Fnd A.) 5ytv-2ux B.) 5yY -2ux Y -2u c.) 2x +Fy" 0.) (2 ++F3)(v-u E.) 2x+ Sy"
4) Let f(x,y, z ) = . In(r+2y + 3z). Find Vf =
If f(x, y,z)= 1 ôf then х-у-32 ôy
Q4 * If f(z) = z? ;z = x+ iy Show me u(x, y);v(x, y) Determine ux; vy|
Evaluate f(x, y, z) dS. f(x, y, z) = x² + y2 + z2 S: x2 + y2 36, 0 SxS 6, 0 SzSx !!
Find f,(1, 1) f(x, y) = In(- Vx2 + y2