Differentiation and turning points problem

Here is what I get:

f = (x+1)^2
g = (1+x^2)

(f/g)' = (f'g - fg') / g^2

= [ (2(x+1)(1+x^2) - (x+1)^2 (2x) ] / (1+x^2)^2

That will be zero only if the numerator is zero so

2(x+1)(1+x^2) = 2x(x+1)^2

x=-1 is one solution right off the bat. Assume that there is another and divide both sides by 2(x+1)

1+ x^2 = x(x+1)

Simplify and x = 1

You should have local extremes at 1 and -1.

I think when you took your derivative and simplified the numerator that you flipped the sign on your x^2 term. It simplifies down to

2(-x^2 + 1) / (x^2+1)^2

I redid the derivative and ended up getting:

(-2x^2 + 1) / (x^2 + 1)^2

which is what u got.

I found the second derivative to be:

(-x^5 - 2x^3 - x^2 - x) / (x^2 + 1)^4

Im not sure if i got the negative sign around the right way somewhere in my working, which may have added terms instead of them cancelling down, just looking for a bit of verification.

Thanks for the previous help, hope to hear from u again.

I'm not sure what you are after with the second derivative. Are you trying to determine if the shape is concave up or down?

Thats is one part of it, but also to find the possible points of inflection when d^2y/ dx^2 or the second derivative is equal to zero. So i need to find the second derivative, at least i think i do.

so did i get the right second derivative?

You have a small difference between what I got and what you got for the first derivative. I got:

2(-x^2 + 1) / (x^2+1)^2

or

-2 (x^2 - 1) / (x^2 +1)^2

The derivative of that is:

-2 [ 2x(x^2+1)^2 - 4x(x^2 -1)(x^2+1)] / (x^2+1)^4

which looks similar to what you got if you expand the terms. If you want to solve this for zero, you can ignore the -2 in the front and the demoninator completely. You get:

2x(x^2+1)^2 = 4x(x^2-1)(x^2+1)

x=0 is clearly a solution. You can simplify this to

(x^2 + 1) = 2 (x^2 - 1)
x^2 = 3
x= +/- sqrt(3)

Your three inflection points are zero, sqrt(3) and -sqrt(3) assuming I didn't make any errors.

that small error was a transference thing, my bad, it was actually + 2 in the expansion.

Shouldn't the 2nd derivative that u show be

-2[ 2x(x^2+1)^2 + 2x(x^2-1)(x^2+1)] / (x^2+1)^4

yours was:

-2 [ 2x(x^2+1)^2 - 4x(x^2 -1)(x^2+1)] / (x^2+1)^4

which comes from

-4x(x^2+1)^2 - (-2x^2+2)(4x(x^2+1)) / (x^2+1)^4

From what i see it looks you have applied the -2 to:
-4x(x^2+1)^2 - (-2x^2+2)
but not:
(4x(x^2+1)

I am probably just saying something wrong, but if so could u explain what i dont get here?

Thanx for all your help.

I'll write it all out and maybe we can see the differences. Starting from here:

-2 (x^2 - 1) / (x^2 +1)^2

Let
-2 = a constant we will carry forward
f = (x^2-1)
g = (x^2 + 1)^2

Then
f' = 2x
g' = 2(x^2+1)(2x) = 4x(x^2+1)

(-2)(f/g)' = (-2)(f'g - fg') / g^2
= (-2) [2x(x^2+1)^2 - (x^2-1)(4x)(x^2+1)] / (x^2+1)^4

Does that look correct?

okay i see

Thanx heaps