   Chapter 14.8, Problem 50E

Chapter
Section
Textbook Problem

(a) Maximize ∑ i   =   1 n x i   y i subject to the constraints ∑ i   =   1 n x i 2   =   1 and ∑ i   =   1 n y i 2   =   1 .(b) Put x i   =   a i ∑   a j 2   and   y i   =   b i ∑   b j 2   to show that ∑ a i   b i   ≤   ∑   a j 2   ∑   b j 2   for any numbers a1, . . . , an, b1, . . . , bn. This inequality is known as the Cauchy-Schwarz Inequality.

(a)

To determine

To find: The maximum value of i=1nxiyi subject to the constraints i=1nxi2=1 and i=1nyi2=1 .

Explanation

Given:

The given function is i=1nxiyi subject to the constraints i=1nxi2=1 and i=1nyi2=1 .

Calculation:

The given function is f=i=1nxiyi , g=i=1nxi21 and h=i=1nyi21 .

The Lagrange multipliers f=λg+μh is computed as follows,

f=λg+μh(i=1nxiyi)=λ(i=1nxi21)+μ(i=1nyi21)y1,...,yn,x1,...,xn=λ2x1,...,2xn,0,...,0+μ0,...,0,2y1,...,2yn

Thus, the value of f=λg+μh is,

y1,...,yn,x1,...,xn=λ2x1,...,2xn,0,...,0+μ0,...,0,2y1,...,2yn .

The result, y1,...,yn,x1,...,xn=λ2x1,...,2xn,0,...,0+μ0,...,0,2y1,...,2yn can be expressed as follows,

yi=2λxi,xi=2μyi for all 1in .

Substitute yi=2λxi in i=1nyi2=1 ,

i=1n(2λxi)2=1i=1n4λ2xi2=14λ2i=1nxi2=14λ2=1

Simplify further as follows.

λ2=14λ=±12

Substitute λ=12 in yi=2λxi ,

yi=2(12)xiyi=xi

For all 1in , yi=xi

(b)

To determine

To show: The aibiaj2bj2 where a1,...,an,b1,...,bn be any numbers.

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 1520, simplify the expression. 17. 16x5yz481xyz54; x 0, y 0, z 0

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Problems 15-20, find the indicated derivative. 20. Find the second derivative

Mathematical Applications for the Management, Life, and Social Sciences

The graph of is: a) b) c) d)

Study Guide for Stewart's Multivariable Calculus, 8th

sinx=ex+ex2

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 