Find the intervals in which the function f given by `f(x)=(4sinx-2x-x cox)/(2+cos x)` is (i) increasing (ii) decreasing.

Explanation

`y = (4 sin x)/(2+ cos x ) - x`
` rArr (dy)/(dx) = ((2+cos x)*4 cos x - 4 sin x * (-sin x))/((2 + cos x)^(2)) - 1`
` = (8 cos x + 4 cos^(2) x + 4 sin^(2) x)/((2+ cos x )^(2)) -1`
` = (4+ 8 cos x )/((2+ cos x)^(2))-1= (4+8 cos x - (2+cos x )^(2))/((2+ cos x )^(2))`
` = (4 cos x - cos^(2)x)/(2+ cos x ) = (cos x ( 4- cos x ))/((2+ cos x )^(2))`
Now ` (dy)/(dx) ge 0 `
` rArr (cos x (4- cos x ))/((2+ cos x )^(2)) ge 0 `
` rArr cos x ge 0 " "(.:' 4 - cos x gt 0)`
`rArr x in [ 0, pi/2 ] `
Therefore, the function is increasing in `[0, pi/2]`.