If `4x+8cos x+tanx-2 sec x-4 log {cos x(1+sinx)}ge 6` for all `x in [0, lambda)` then the largest value of `lambda` is

Options

A. `pi//3`
B. `pi//6`
C. `pi//4`
D. `3pi//4`

Answer

B

Explanation

Let `f(x)=4x+8 cos x +tan x-2 sec x`
`-4log {cos x(1+sin x)}-6`
Now `f(0)=0+8+0-2-4log1-6=0 and`
`f'(x)=4-8sinx+sec^(2)x-2 sec x tan x+4 tanx+4 tan x-(4 cos x)/(1+sinx)`
`=4(1-2sinx)+sec^(2)x(1-2sinx)+(4sinx)/(cosx)-(4cosx[1-sinx])/(cos^(2)x)`
`=(4+sec^(2)x)(1-2 sin x)+(4)/(cosx)(sinx-1+sinx)`
`=(1-2sinx)(4+sec^(2)x-4 secx)`
`=(1-2 sinx)(secx-2)^(2)`
Thus, `f'(x)gt0` fo