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Let nge1, n in Z. The real number a in 0...

Let `nge1, n in Z`. The real number `a in 0,1` that minimizes the integral `int_(0)^(1)|x^(n)-a^(n)|dx` is

A

`(1)/(2)`

B

2

C

1

D

`(1)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `I(a)=int_(0)^(1)|x^(n)-a^(n)|dx`
`therefore" "I_(a)=int_(0)^(a)(a^(n)-x^(n))dx+int_(a)^(1)(x^(n)-a^(n))dx=(2n)/(n+1).a^(n+1)-a^(n)+(1)/(n+1)`
`rArr" "(d)/(da)(I(a))=n(2a-1)a^(n-1)rArr " only critical point of in (0, 1) is a"in((1)/(2),1),I(a)" is minimum for a"=(1)/(2).`
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