Question

Show that `f(x) =(4 sinx)/(2+cosx) -x` is an increasing function of x in `[0, pi/2]`

Answer 1

`f(x) =(4 sinx)/(2+cosx) -x`

d.w.r to x

`f'(x) = ((2+cosx) d/dx (4sinx)-(4sinx) d/dx (2+cosx))/(2+cos x)^2 - dx/dx`

`= ((2+cosx)(4cosx)-(4sinx)(0-sinx))/(2+cos)^2 -1`

`= (8cosx+4cos^2x+4sin^2x)/(2+cos)^2 -1`

`= (8cosx+4(cos^2x+sin^2x))/(2+cosx)^2 -1`

`=(8 cos x + 4(1))/(2+cosx)^2-1`

`= (8cosx+4-4cos^2x-4cosx)/(2+cosx)^2`

`= (4cosx-cos^2x)/(2+cosx)^2`

`= (cosx(4-cosx))/(2+cos x)^2`

Here, (2 + cos x)² > 0

`cosx≥0, x ∈ [0, pi/2]`

Hence, `f(x) = (4sinx)/(2+cosx) - x` is an increasing function of x in `[0, pi/2]`