(a) Explanation Given information: The function f ( x ) is f ( x ) = ( a 1 x + b 1 ) 2 + ( a 2 x + b 2 ) 2 + ⋅ ⋅ ⋅ + ( a n x + b n ) 2 . Calculation: Provide the function as shown below: f ( x ) = ( a 1 x + b 1 ) 2 + ( a 2 x + b 2 ) 2 + ⋅ ⋅ ⋅ + ( a n x + b n ) 2 (1) The function f ( x ) = ( a 1 x + b 1 ) 2 is a quadratic function of the form ( a 1 x ) 2 + ( b 1 ) 2 + 2 ( a 1 x ( b 1 ) ) with coefficient of a = ∑ a 1 2 , b = ∑ a 1 b 1 , c = ∑ b 1 2 . Expand the equation (1). f ( x ) = ( a 1 x + b 1 ) 2 + ( a 2 x + b 2 ) 2 + ⋅ ⋅ ⋅ + ( a n x + b n ) 2 = ( a 1 2 x 2 + 2 a 1 b 1 x + b 1 2 ) + ... + ( a n 2 x 2 + 2 a n b n x + b n 2 ) = ( a 1 2 + a 2 2 + (b) To determine Prove that the equality holds in the Cauchy-Schwarz inequality provided there exists a real number x which makes a i x equal b i for each value i from 1 to n .

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ISBN: 9780134175706

Author: Unknown

Publisher: PEARSON

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Question

Chapter 4, Problem 34AAE

(a)

To determine

Deduce the Cauchy-Schwarz inequality.

(b)

To determine

Prove that the equality holds in the Cauchy-Schwarz inequality provided there exists a real number x which makes aix equal bi for each value i from 1 to n.

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Chapter 4, Problem 33AAE

Chapter 4, Problem 35AAE

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