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

Options

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

Answer

B, C

Explanation

We have
`f(x)=|{:(cos2x" "cos2x" "sin 2x),(-cosx" " cos x " " -sinx),(sinx " "sinx " "cos x):}|`
`rArr f(x)=|{:(cos2x" "0" "sin 2x),(-cosx" " 2cos x " " -sinx),(sinx " "0 " "cos x):}|` [ Applying `C_2rarr C_2 - C_1]`
`rArr f(x)=2cosx|{:(cos2x " "sin 2x ),(sin x " "cos x):}|= 2 cos x cos 3x `
`rArr f(x) = cos 4x + cos2 x)`
`rArr f(x)=-4 sin 4x -2 sin 2x =-2 sin 2x (1+4 cos 2x)`
`therefore ` f(x) =0
`rArr -2 sin 2x (1+ cos 2x)=0`
`rArr sin2 x=0 or cos 2x =-1/4`
Now `2x=-1/4` has four solutions in`( -pi ,pi) as y= 1/4` intersects y= cos 2x at four points in `(-pi ,pi)`
Now f(x) =-4 sin 4x -4 cos 2x
`rArr f(0) =-20 lt 0 `
Thus (x) attains maximum at x=0