Graph/analyze a function of a rational /w complex root?

[Saterial]

graph/analyze a function of a rational /w complex root?!?

Homework Statement

The function is y=x^2/(x^2+3)

Homework Equations

First and Second Derivatives
Chart to find intervals of increase/decrease and concavity.

The Attempt at a Solution

1) Domain
{XeR}
2) Intercepts
If x=0, y=0
(0,0) is the only intercept.
3) Symmetry
(-x,y) = (x,y) therefore, even.
4) Vertical Asymptotes
My question here is, since the "asymptote" would technically be the sqroot of 3i, in my class we don't use complex numbers/roots so would I simply answer this as "does not exist" or saw it equals 0?
5) Horizontal Asymptote
lim x->+- infinity 1
y=1
6) Slant Asymptotes
none
7) First Derivative
dy/dx = 6x/(x^2+3)^2 (quotient rule)
8) Second Derivative
d2y/dx2 = -18(x+1)(x-1)/(x^2+3)^3 (quotient rule + chain rule)
9) Critical numbers
Set first deriv equal to 0, critical number is x=0
10) Intervals of Increase/Decrease
This is where I start having trouble. Now I know that in order to find the values of increase and decrease, you need to use your critical numbers and restrictions. The method given to use to find the intervals is to plug it into a chart and see at which intervals is the graph positive or negative. However, my question is, what would the intervals I test be? Since my restriction is a complex root and we don't work with those in our class; what would I use for my intervals? Simply (-infinity, 0), (0, infinity) ? and my values I would be testing for are 6x, and (x^2+3)^2 ?
11) Concavity
Set second deriv = to 0. Would my x values be +-1 and again, would I use anything for restriction, use 0 as an interval ? or?
12) Intervals of Concavity
This, I also have a similar issue to finding just like intervals of increase/decrease, what intervals would I be using? Do I include anything in intervals that relates to restrictions? or do I simply use (-infinity,-1),(-1,1) and (1,infinity) ? and the binomials/intervals I would be testing these in are, -(x+1), -(x-1), and (x^2+3)^3?

Any help would GREATLY be appreciated, I can't figure out this root issue.

 

[micromass]

4) Vertical Asymptotes
My question here is, since the "asymptote" would technically be the sqroot of 3i, in my class we don't use complex numbers/roots so would I simply answer this as "does not exist" or saw it equals 0?

You are right; The asymptote would be at 3i and -3i. But we only work with real numbers. And since there are no real asymptotes, we just say that the asymptote does not exist.

10) Intervals of Increase/Decrease
This is where I start having trouble. Now I know that in order to find the values of increase and decrease, you need to use your critical numbers and restrictions. The method given to use to find the intervals is to plug it into a chart and see at which intervals is the graph positive or negative. However, my question is, what would the intervals I test be? Since my restriction is a complex root and we don't work with those in our class; what would I use for my intervals? Simply (-infinity, 0), (0, infinity) ? and my values I would be testing for are 6x, and (x^2+3)^2 ?

Correct again. You split up your domain in (-infinity,0) and (0,infinity). And on these two pieces you check whether your first derivative is negative/positive.

11) Concavity
Set second deriv = to 0. Would my x values be +-1 and again, would I use anything for restriction, use 0 as an interval ? or?

Your x values are +-1.

12) Intervals of Concavity
This, I also have a similar issue to finding just like intervals of increase/decrease, what intervals would I be using? Do I include anything in intervals that relates to restrictions? or do I simply use (-infinity,-1),(-1,1) and (1,infinity) ? and the binomials/intervals I would be testing these in are, -(x+1), -(x-1), and (x^2+3)^3?

Correct again, you split up the domain in (-infinity,-1), (-1,1) and (1,infinity). Then in each of these parts you check whether your second derivative is negative/positive.