Question

Concavity and Inflection (Medal to the winner. Comments...)

Answer 1

I will give you medal if you show me how to solve this problem, XD

Answer 2

That is why I posted it.

Answer 3

First find first and the second derivative.

Answer 4

After I get the second?

Answer 5

I know f''(x)=-12x^2-36x=0

Answer 6

Once you have the second, find where it's equal to zero, those are your points where it changes concavity

Answer 7

But how do I denote that?
X=-3
X=0

Answer 8

Now you have to find where it's concave up and down between those intervals. You could pick random points between them, and find f''

Answer 9

When f'' is positive = concave up

f'' negative = concave down

Answer 10

So I choose any number between -3 and 0?

Answer 11

And you want to check outside the interval. The function changes concavity twice.

Answer 12

Alright, so if I chose -2 (f(x)=24) and 3 (f(x)=(-216) is that alright since one is negative and the other is positive?

Answer 13

Still need to check the other side of -3, though. But yes that tells you it's concave up where -3<x<0, and concave down x>0

Answer 14

I'm sorry, how do I check the other side of -3?

Answer 15

The same way you checked the other side of 0 :P pick a number less than -3

Answer 16

I'm sorry xD I'm running out of time with the problem and I don't understand how they want me to denote it in the end.

Answer 17

-3<x<0 is the same as (-3, 0). x>0 means (0, inf)

Answer 18

I never knew that.

Answer 19

How do I get inflection points then?

Answer 20

You already found them lol...

Answer 21

The first things you found!

Answer 22

Oooooooooooookaaaaaaaaaay I seeeeee...so my final answer would be
CU: (-3,0)
CD: (0,inf)
IP: (-3,0)?
Sorry if I am getting this wrong, I'm learning this myself. My teacher hasn't been available and his substitute is less than accommodating :S

Answer 23

Never mind, I see it is wrong with the last two ones.

Answer 24

Alright, so I got CU, and IP (they wanted it without the parenthesis)...now to get that CD...thank you for your help thus far by the way...a lot.

Answer 25

Alright, I picked -4 for the CD, and got -48.

Answer 26

CD is two intervals (-inf, -3) as well