Need help finding intervals where the function is decreasing.

Question

asked Apr 25, 2020 10.0k views

1 Answer

Alex Calugarescu · Answer 1 · 2020-05-01T20:52:38+0000

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

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

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