Question

Consider the equation below.
f(x) = 3 cos^2x − 6 sin x, 0 ≤ x ≤ 2π
(a) Find the interval on which f is increasing. (Enter your answer in interval notation.)
b)Find the interval on which f is decreasing. (Enter your answer in interval notation.)
c)(b) Find the local minimum and maximum values of f.
d)(c) Find the inflection points.
e)Find the interval on which f is concave up. (Enter your answer in interval notation.)
f)Find the interval on which f is concave down. (Enter your answer in interval notation.)
Please help me work this out!

Answer 1

\[f(x)=3\cos^2(x)-6\sin(x)\] right?

Answer 2

yes

Answer 3

did you take the derivative as a first step?

Answer 4

and if so, what did you get?

Answer 5

-6cos(x)sin(x)-6cos(x)

Answer 6

that looks good
i guess the next step is to factor

Answer 7

you got that?

Answer 8

once you factor out the \(-6\cos(x)\) it is going to be real easy to see where the derivative is positive and negative

Answer 9

-6cos(x)(sin(x)+1)?

Answer 10

yeah and forturnately for you, since \(\sin(x)\geq -1\) always, you know \(\sin(x)+1\geq 0\) so you can ignore that part when you check tor the sign of the derivative

Answer 11

in other words, the sign of the derivative is completely dependent on the sign of \(-6\cos(x)\)

Answer 12

you got that?

Answer 13

no, im confused as to what my answer would be

Answer 14

ok lets go slow

Answer 15

it is clear that in order to find the interval over which the function is increasing (decreasing) your only job is to find the interval over which the derivative is positive (negative) yes?

Answer 16

yes

Answer 17

ok and your derivative isi
\[-6\cos(x)\left(\sin(x)+1\right)\]

Answer 18

so the question is, over what interval is the derivative positive

Answer 19

so i would have to graph it to see it clearly right?

Answer 20

you have two factors \[-6\cos(x)\] and \[\sin(x)+1\]
no you do not have to graph it

Answer 21

is it clear that
\[\sin(x)+1\geq 0\] for any value of \(x\)?

Answer 22

yes

Answer 23

that means when you are trying to find if the derivative is positive or negative, you can ignore that factor entirely (because it is never negative) and concentrate only on \(-6\cos(x)\)

Answer 24

do you know over what intervals between \(0\) and \(2\pi\) that cosine is positive?

Answer 25

i cant remember

Answer 26

then i guess i have to tell you

Answer 27

it is positive on \((0,\frac{\pi}{2})\) that is on the right sides of the unit circle

Answer 28

negative on \((\frac{\pi}{2},\frac{3\pi}{2})\) that is the left side

Answer 29

and positive again on \((\frac{3\pi}{2},2\pi)\)

Answer 30

since your cosine has a \(-6\) in front of it, it will be negative where cosine is positive, and vice versa

Answer 31

okay so for a) it would be pi/2,3pi/2 ?

Answer 32

increasing on that interval, yes

Answer 33

and of course decreasing on the remaining intervals

Answer 34

to find the max and min i would plug in to the equation pi/2 and then 3pi/2?

Answer 35

yes

Answer 36

in to the original function of course, not the derivative

Answer 37

got 6 and -6

Answer 38

looks good

Answer 39

thank you so much!

Answer 40

yw