Which of the following is incorrect?

Options

A. `(pi)^(1//pi)lt(e)^(1//e)`
B. `(10)^(1//10)gt(11)^(1//11)`
C. `(In2)^(1//(In2))lt(In3)^(1//(In3))`
D. `(tan^(-1)3)^(1//(tan^(-1))3)lt(tan^(-1)2)^(1//(tan^(-1)2))`

Answer

D

Explanation

take `f(x) = x^(1//x)`
`log f(x) = (log^(x))/(x) " " rArr" " (f(x))/(f(x)) = (1-Inx)/(x^(2))`

`f.(x) = ((1-In x)/(x^(2))) . x^((1)/(X))`
For (x) is increasing for `0 lt xlt e`
And decreasing for `x gt e`
`pi^((1)/(pi)) lt e^((1)/(e))" " because pi gt e & f (x) darr`
`10^((1)/(10)) lt 11^((1)/(11) " " because 11 gt 10 gt e ," " f(x) darr`
The function is increasing for `x lt e` , therefore `f(In 2) lt f (In 3) & f(tan^(-1) 2) lt f( tan^(-1)3)`