Which of the following is incorrect :

Options

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

Answer

D

Explanation

take , `f(x) = x^(1//2)`
`log f(x) = (log^(x))/(x) " " rArr " "(f(x))/(f(x)) = (1- In x)//(x^(2))`
`f'(x) = ((1- In x)/(x^(2)) . X^((1)/(X))`
F(x) is increasing for `0 lt x lt e `
And decreasing for ` x gt e`

`(1)/(pi^(x)) lt (1)/(e^(e))" " becaue 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)`.