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The fraction exceeds its p^(th) power b...

The fraction exceeds its `p^(th) ` power by the greatest number possible, where `p geq2` is

A

`((1)/(p))^(1//(p-1))`

B

`((1)/(p))^(p-1)`

C

`p^(1//p-1)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
Let `y=x-x^(p)`, where x is the fraction
`rArr" "(dy)/(dx)=1-px^(p-1)`
For maximum of minimum, `(dy)/(dx)=0`
`rArr" "1-px^(p-1)=0 rArrx=((1)/(p))^(1//(p-1))`
Now, `(d^(2)y)/(dx^(2))=-p(p-1)x^(p-2)`
`therefore(d^(2)y)/(dx^(2)):|_(x=((1)/(p))^(1//(p-1)))=-p((1)/(p))^((p-2)//(p-1))lt0`
`therefore "y is maximum at x"=((1)/(p))^(1//(p-1))`
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