If `f (x) = sqrt(cos ec ^(2) x - 2 sin x cos x - (1)/(tan ^(2) x ))`
`x in ((7pi)/(4), 2pi )` then ` f' ((11 pi)/(6))=`

Options

A. `(1- sqrt3)/(2)`
B. `(sqrt3-1)/(2)`
C. `(1+sqrt3)/(2)`
D. None of these

Answer

A

Explanation

`f (x) = sqrt(cosec ^(2) x - cot ^(2) x-2 sin x cos x )`
`= sqrt)(cos x-sin x )^(2))`

`f (x)= cos x -sin x`
`implies f'(x) =-sin x - cos x`
`f'((11pi)/(6))=-[sin ""(11pi)/(6)+ cos ""(11pi)/(6)]`
`implies f' ((11pi)/(6))=-[(-1)/(2) + (sqrt3)/(2) ] = (1- sqrt3)/(2)`