   Chapter 4.5, Problem 104E

Chapter
Section
Textbook Problem

PUTNAM EXAM CHALLENGEFind all the continuous positive functions f(x), for 0 ≤ x ≤ 1 , such that ∫ 0 1 f ( x )   d x = 1 ∫ 0 1 f ( x ) x   d x = α ∫ 0 1 f ( x ) x 2   d x = α 2 where α is a given real number.

To determine

To calculate: All continuous positive functions f(x), for 0x1 and satisfies the conditions.01f(x)dx=101f(x)xdx=α01f(x)x2dx=α2

Explanation

Given: For, 0x1,

01f(x)dx=101f(x)xdx=α01f(x)x2dx=α2

Formula used:

Calculation:

Multiply α2 to both sides of 01f(x)dx=1 to get,

α201f(x)dx=1(α2)α201f(x)dx=α2

So,

α201f(x)dx=α2 …… (1)

Multiply 2α to both sides of 01f(x)xdx=α to get,

2α01f(x)<

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