   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.

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