[tex]f(x)=-5x-5\sin x\implies f'(x)=-5-5\cos x[/tex]
[tex]f[/tex] is decreasing on some interval [tex](a,b)[/tex] if [tex]f'(x)<0[/tex] for all [tex]x\in(a,b)[/tex]. Similarly, [tex]f[/tex] is increasing on [tex](a,b)[/tex] if [tex]f'(x)>0[/tex] for all [tex]x\in(a,b)[/tex].
On the domain [tex]0\le x\le2\pi[/tex], we have
[tex]-5-5\cos x=0\implies\cos x=-1\implies x=\pi[/tex]
as the only critical point. So we need to check the sign of [tex]f'[/tex] on two intervals, [tex](0,\pi)[/tex] and [tex](\pi,2\pi)[/tex].
On [tex](0,\pi)[/tex], pick any test value. Suppose we take [tex]x=\dfrac\pi2[/tex]. Then [tex]f'\left(\dfrac\pi2\right)=-5<0[/tex], so [tex]f[/tex] is decreasing on this interval.
On [tex](\pi,2\pi)[/tex], you'd find that [tex]f'>0[/tex], so [tex]f[/tex] is increasing on this interval.
