Chapter 5.4, Problem 115E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# PUTNAM EXAM CHALLENGEFor each continuous function f: [ 0 ,   1 ] → ℝ , let I ( f ) = ∫ 0 1 x 2 f ( x ) d x and J ( x ) = ∫ 0 1 x ( f ( x ) ) 2 d x . Find the maximum value of I ( f ) − J ( f ) over all such functions f.This problem was composed by the Committee on the Putnam Prize Competition. © The Mathematical Association of America. All rights reserved.

To determine

To calculate: The maximum value of I(f)J(f) if I(f)=01x2f(x)dx and J(f)=01x(f(x))2dx

Explanation

Given:

The integrals are

I(f)=âˆ«01x2f(x)dx And J(f)=âˆ«01x(f(x))2dx

For each continuous function f:[0,1]

Formula used:

a2+b2âˆ’2ab=(aâˆ’b)2

The integration is done according to

âˆ«xndx=xn+1n+1

Calculation:

The integrals given are I(f)=âˆ«01x2f(x)dx and J(x)=âˆ«01x(f(x))2dx

Subtract I(f)âˆ’J(f),

I(f)âˆ’J(fâ€Š)=âˆ«01x2f(x)dxâˆ’âˆ«01x(f(x))2dx=âˆ«01x2f(x)âˆ’x(f(x))2dx â€¦...â€¦... (1)

Add and subtract x34, to get

x2f(x)âˆ’x(f(x))2=x34âˆ’xf(x)2+x2f(x)âˆ’x34=x34âˆ’x(f(x)2&

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