You are given the following(X^2)/(X^2+3)FIND when f is decreasing, increasing, concave up, down, and inf point?

find the first derivative by calculator, which is (6x)/((x^2+3)^2) then equal it to zero......x=0

check numbers in the derived equation as -1 which is dec and check after such as +1 which is inc ....

the same to find second derivative by calculator....then equal it to zero....the points are -1 & +1....then of course we check between these numbers such as -2, 0 , 2.....which concave down (-inf,-1) & (1,inf)

concave up (-1, 1)...

inflection point : plug the numbers -1 & +1 into the original equation.

if you want to find the max and minimum, additional info, graph the function and the plug the number 0 in the original equation.

so your function is:

F(x)=x^2/(x^2+3)

we need to find all the interesting points

we know that you can't divide by 0 so when the denominator is 0 is an interesting point

0=x^2+3

x=+/- i sqrt(3)

it's interesting but since it's not in the real plane it won't show up as a asymptote

another interesting point would be the root when F(x)=0

0=x^2/(x^2+3)

0=x^2

there's a double root at x=0

looking at a plot this should show up as a "bounce" off the x axis at 0

the next interesting points would be critical points or when F'(x)=0

F'(x)=(2x(x^2+3)-x^2*2x)/(x^2+3)^2

simplified

F'(x)=6x/(x^2+3)^2

the denominator still can't ever be negative because of the square and +3 so the critical points can be found at

0=6x

simplified

x=0 which makes sense since we had a double root at x=0

so it will either be increasing or decreasing from

(-infin,0) and (0,infin)

the second derivative will tell which of two choices it will be

so

F''(x)=(6*(x^2+3)^2-6x*2*(x^2+3)*2x)/(x^2+3)^4

simplified:

F''(x)=(6*x^4+36x^2+54-24x^4-72x^2)/(x^2+3)^4

F''(x)=(-18x^4-36x^2+54)/(x^2+3)^4

when you plug in the critical point x=0 into F''(x)

a negative number would mean it's going from increasing to decreasing

a positive number would mean it's going from decreasing to increasing

a 0 would mean it's a concavity point and that it's not changing so you would have to plug a value from each domain and see if it's positive or negative and that will tell you if it's increasing or decreasing respectively

F''(0)=.667 which is >0 so

it's decreasing from (-infin,0) and increasing from (0,infin)

to find the concavity set F''(x)=0

0=(-18x^4-36x^2+54)/(x^2+3)^4

still nothing going on with the denominator so:

0=-18x^4-36x^2+54

to find the roots you can pretend each x^2 is a u so it will look

0=-18u^2-36u+54

which is solvable using the quadratic equation

u=(36+/-sqrt(36^2-4*-18*54))/(2*-18)

which gives you

u=-3; and u=1; (i guess you could have factored it but this worked so yeah)

so the domains you need to find the concavity are from

(-infin,-3),(-3,1),(1,infin)

you could plug each of the values in and see whether or not it's positive or negative and find the concavity that way

or you could just look at a nice graph like this:

http://www.wolframalpha.com/input/?i=x^2/(x^2%2B3)

it makes a U near 0 so that would be concave up so the other two domains would be concave down

it seems i made a mistake in my math somewhere because it should be symmetric

but i'm out of time so you can go find it :D