Question

Calculus concavity?

determine the open intervals on which the
graph is concave upward or concave downward

Answer 1

Derivatives... can you find them?

Answer 2

derviative would be f'(x)=6x-3x^2

Answer 3

is that right?

Answer 4

That's the first derivative... where it's positive, it tells you when the function is increasing, and where it's negative, that's where the function is decreasing... however, that's not quite what we want to know, now, is it? ^_^

Find the second derivative...

Answer 5

You still there?
\[\Large f''(x) = \color{red}?\]

Answer 6

yeah sorry

Answer 7

not sure.... is it 6-6x

Answer 8

As a matter of fact, yes :D
Now, study that second derivative...
in the open interval where it is positive, the graph is concave upward, in the open intervals where it is negative, the graph is concave downward....

Answer 9

so i need to find the open intervals of both the first and second derivative?

Answer 10

NO, just the second ^_^
You only use the first derivative if you want to find where it is increasing/decreasing.

Answer 11

how do I do that?????

Answer 12

Let's look for the interval where it is concave downward first, shall we?
For that, we need the second derivative to be negative...
\[\Large 6-6x < 0\]
solve for x.

Answer 13

-6x<0

Answer 14

sorry

Answer 15

i meant -6x<-6

Answer 16

-6x/-6<-6/-6

Answer 17

x=1

Answer 18

It's not an equals-sign...

Answer 19

x<1

Answer 20

yes?

Answer 21

@terenzreignz

Answer 22

oh my goodness. thank you so much. I really appreciate it