If `f(x) = |{:( cos (2x) ,, cos ( 2x ) ,, sin ( 2x) ), ( - cos x,, cosx ,, - sin x ), ( sinx,, sin x,, cos x ):}|`, then

Options

A. `f(x) ` attains its minimum at `x =0`
B. ` f(x)` attains its maximum at ` x =0`
C. `f' (x)=0` at more than three points in `(-pi , pi )`
D. `f' (x) =0` at exactly three points in ` (-pi, pi )`

Answer

B, C

Explanation

`f(x) = |{:( cos 2x,, cos 2x,, sin 2 x ), ( -cos x,, cos x,, - sin x ), ( sin x,, sin x,, cosx):}| `
` cos 2x ( cos ^(2) x + sin ^(2) x) - cos 2x `
`" " (- cos ^(2)x + sin ^(2) x ) + sin 2x (- sin 2x ) `
`" " = cos 2x + cos 4x `
` f ' (x) = - 2sin 2x - 4 sin 4x = - 2 sin 2x ( 1+ 4 cos 2x ) `
At `" " x =0`
` f'(x) = 0`
and ` f (x) = 2 `
Also, ` f ' (x) = 0`
` sin 2x = 0`
or ` cos 2x = (-1)/(4) `
`rArr " " x = ( n pi )/(2) or cos 2x = - (1)/(4)`