Question

please help to find
critical and increasing and decreasing intervals,
i am really confused!

Answer 1

Hi there! Hopefully, I can help you out and not confuse you. :D First all of do you know how to find critical numbers of a function at all?

Answer 2

slightly yes

Answer 3

Give a minute. :) Working out the derivative of that function.

Answer 4

Okay, well, what do you know so far?

Answer 5

it should be 56x/(x^2+4)^2

Answer 6

Okay, I figured out the derivative and the critical numbers of the function. Yeah, got that, too.

Answer 7

Basically to find out the intervals where a function is increasing, you're going to use any critical numbers -- where f'(x) = 0 or any numbers where the function undefined to make test intervals. Does that make sense?

Answer 8

so first I have to find ciritcal points. How do i do that?

Answer 9

Okay. Basically to find critical points, you are going to set f'(x) = 0 and solve for x. That will give you your critical points.

Answer 10

so it would be 56x/(x^2+4)^4=0?

Answer 11

what shall i do next step?

Answer 12

\[f'(x) = \frac{ 56x }{ (x^2+2)^2 }\]\[\frac{ 56x }{ (x^2+4)^2 } = 0\]\[(x^2+4)^2*\frac{ 56x }{ (x^2+4)^2 } = 0*(x^2+4)^2\]
\[56x = 0\]
\[x = 0\]

Answer 13

thats cool.
so ciritcal point is zero.?

Answer 14

So, when can use x = 0 to divide the domain of f(x) and test the f'(x) for values of x on those intervals.

Answer 15

Yes. :)

Answer 16

In the case of this function, the domain is:\[(-\infty, \infty)\]

So, you would use x = 0 to divide that into two intervals.

\[(-\infty, 0), (0, \infty)\]

Answer 17

With me so far?

Answer 18

ook, when you are talking about domain, you are taking about domain of f(x)'?

Answer 19

Yes. The domain of the original function, f(x).

Answer 20

origional function, ok.

Answer 21

sorry I have to go, but I will be back around 45 mins.

Answer 22

No problem. Sorry for taking so long. I'll put up what I have, though.

Answer 23

Basically what you are going to do next, is pick values of x within your intervals and test f'(x). You're going to evaluate f'(x) at the values you have chosen. The sign on derivative is going to tell if the function is increasing or decreasing on that interval.

If the sign of f'(x) is negative, then the function f(x) is decreasing on that interval.
If the sign of f'(x) is positive, then the function f(x) is increasing on that interval.

Answer 24

I've attached a table showing what I did, as well as the graph of the function to verify that my answers. I hope that it helps. If you have need more help let me know. :)