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 $f'(x)$ 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 $x=\dfrac\pi2$. Then $f'\left(\dfrac\pi2\right)=-5$, so $f$ is decreasing on this interval.

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