   Chapter 14, Problem 8P

Chapter
Section
Textbook Problem

Show that the maximum value of the function f ( x , y ) = ( a x + b y + c ) 2 x 2 + y 2 + 1 is a2 + b2 + c2. Hint: One method for attacking this problem is to use the Cauchy-Schwarz Inequality: | a ⋅ b | ≤ | a | | b |

To determine

To show: Maximum value of the function f(x,y)=(ax+by+c)2x2+y2+1 is a2+b2+c2 .

Explanation

Given:

The given function is f(x,y)=(ax+by+c)2x2+y2+1 (1)

Calculation:

Let, u=(a,b,c) and v=(x,y,1) .

|u|=a2+b2+c2|v|=x2+y2+12u.v=ax+by+c

Write the formula for Cauchy-Schwarz Inequality.

|u.v||u||v| (2)

Substitute a2+b2+c2 for |u| , x2+y2+12 for |v| , and ax+by+c for u.v in

equation (2).

ax+by+c(a2+b2+c2)(x2+y2+12)

Square on both sides,

(ax+by+c)2(a2+b2+c2)(x2+y2+1)(ax+by+c)2x2+y

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