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

Options

A. `(1/n)^(1/(n-1))`
B. `(1/n)^(n-1)`
C. `n^(1/n-1)`
D. non of these

Answer

A

Explanation

Let x be the fraction and let `y=x-x^n` . Then
`dy/dx=1-nx^(n-1)and (d^(2)y)/(dx^2)=-n(n-1)x^(n-2)`
Now `,(dy)/(dx) =0 rarr x= (1/n)^(1/(n-1))`
Clearly `(d^2y)/(dx^2) lt 0 " for " x =(1/n)^(1/(n-1))`
Hence , y is maximum when `x=(1/n)^(n-1)`