Question

Need help finding intervals where the function is decreasing.

Answer 1

`f(x)=-5x-5\sin x\implies f'(x)=-5-5\cos x`

`f` is decreasing on some interval
`(a,b)` if
`f'(x)<0` for all
`x\in(a,b)`. Similarly,
`f` is increasing on
`(a,b)` if
`f'(x)>0` for all
`x\in(a,b)`.

On the domain
`0\le x\le2\pi`, we have

`-5-5\cos x=0\implies\cos x=-1\implies x=\pi`

as the only critical point. So we need to check the sign of
`f'` on two intervals,
`(0,\pi)` and
`(\pi,2\pi)`.

On
`(0,\pi)`, pick any test value. Suppose we take
`x=\frac\pi2`. Then
`f'\left(\frac\pi2\right)=-5<0`, so
`f` is decreasing on this interval.

On
`(\pi,2\pi)`, you'd find that
`f'>0`, so
`f` is increasing on this interval.