Question

Find any critical numbers of the function
(sin(x))^2 + cos(x) 0 < x < 2pi

Answer 1

You just would need to take the derivative and set it equal to 0. Would you know how to do the derivative of this?

Answer 2

That's as far as I can get, haha. I set the derivative to zero then I am stuck. It's the trig throwing me off

Answer 3

Trig sucks for most, haha. Okay, so:

2sinx*cosx - sinx = 0
2sinx*cosx = sinx
2cosx = 1
cosx = (1/2)

Did you make it that far?

Answer 4

Nope. I just made it to 2sinx*cosx = sinx

Answer 5

but I see what you did to get cosx = (1/2)

Answer 6

Yep yep. Recall the two angles where cosx would be 1/2?

Answer 7

Ahh... pi/3? and...

Answer 8

Yep, thats the first one. So, cosx refers to x-values, which would be positive in the 1st and 4th quadrants. So your 2nd angle would be straight down underneath it in the 4th quadrant.

Answer 9

(5pi)/3?

Answer 10

Bingo.

Answer 11

Would that be my answer? I actually (somehow) got that earlier, but it was marked wrong.

Answer 12

Well.....Ill graph it on my calculator to try and confirm the answer. Or confirm we fail, haha

Answer 13

Lol. Ok thanks!

Answer 14

Yeah, I think I can see the problem. There are other values that we have to find by factoring, not by solving like I did. So backtrack:

2sinxcosx - sinx
sinx(2cosx - 1) = 0

sinx = 0
2cosx - 1 = 0

So we only got half of the answers, haha. You need 0 and pi as well.

Answer 15

Ok... so, the answer would be all four? pi/3, (5pi/3), 0, and pi

Answer 16

Yes

Answer 17

It's still marking it wrong, haha :/

Answer 18

It might just be an error in the website?

Answer 19

Nah, lets not assume that just yet.

Answer 20

Nope, im being dumb, drop 0. Remember, interval 0 < x < 2pi, so cant include 0 -_-

Answer 21

YESS!! I'm the dumb one, haha. I thought of that but accidentally just typed 5pi instead of (5pi)/3

Answer 22

Whoops xD

Answer 23

Thank you so much Psymon!

Answer 24

Can I ask you for help on one more, or am I supposed to open a new question?

Answer 25

Go for it.

Answer 26

Nah, you can ask here.

Answer 27

Find any critical numbers of the function.
f(θ) = 8sec θ + 4tan θ, 0 < θ < 2pi

Answer 28

You got the derivative part down?

Answer 29

I got
8secθ*tanθ+4sec^2θ
?

Answer 30

Whoops, im being dumb. Okay, sorry for the delay, I was arguing with myself over something that doesnt matter.

Answer 31

That's ok, haha

Answer 32

So yes, now we factor that derivative answer
8secxtanx + 4sec^2x
4secx(2tanx + secx)
4secx = 0
2tanx + secx = 0

Now for the first one, 4secx = 0
Do you remember much about the values or the graph of secx?

Answer 33

Ah, noo ._.

Answer 34

Ill show ya visually



I dont need to label it properly to show my point. So where does that secant graph ever = 0?