Find the intervals in which the function `f` given by `f(x)=(4sinx-2x-xcosx)/(2+cosx),\ 0lt=xlt=2pi` is increasing and decreasing

Explanation

To find the intervals in which the function \( f(x) = \frac{4 \sin x - 2x - x \cos x}{2 + \cos x} \) is increasing and decreasing over the interval \( 0 < x < 2\pi \), we will follow these steps:

### Step 1: Differentiate the function
We need to find the derivative \( f'(x) \) to determine where the function is increasing or decreasing. We will use the quotient rule for differentiation.

The quotient rule states that if \( f(x) = \frac{u(x)}{v(x)} \), then:
\[
f'(x) = \frac{u'v - uv'}{v^2}
...