Find the interval in which $f(x) = \frac{4\sin x -2x-x\cos x}{2+\cos x}$ increases and decreases.
I differentiated it and got $f'(x)= \frac{4\cos x - \cos^2 x}{(2+\cos x)^2} $. Equating it to $0$ gives $\cos(x) = 0$ i.e. $x = \frac{(2n+1) \pi}{2}, n \in\mathbb{Z}$.
I'm a little confused because of this general form and I can't continue. If given a particular interval like check for increase and decrease in (a, b), I can do it, but this general form is annoying.
I need some help.
If you write $f'(x)= \dfrac{\cos x (4 - \cos x)}{(2+\cos x)^2} $ you can see that the factors $(4 - \cos x)$ and $(2 + \cos x)^2$ are always positive, so $f'(x)$ is positive or negative whenever $\cos x$ is.