Finding Inflection Points: Calculating First and Second Derivatives

[Swerting]

This problem is fairly straightforward. I have a function and domain:
f(x)=2ln(x^{2}+3)-x -3\leqx\leq5
The question eventually asks for the inflection points, which I know occur when f''(x) is zero or undefined.
I found the first derivative and second derivative to be, respectively:
f'(x)=(4x)(x^{2}+3)^{-1}-1
f"(x)=(4)(x^{2}+3)^{-1}-(4x)(x^{2}+3)^{-2}
I have double checked my math and can't seem to find anything wrong with it.
Also, I have looked at a calculator and it shows that the original function does have two inflection points, but when I try to set f"(x) to zero or to be undefined (devide by zero), I always end up with an impossible solution. I don't believe that this involves irrational numbers, so I appreciate any help at all.

 

[HallsofIvy]

Your second derivative is wrong.

You have
f"(x)=(4)(x^2+3)^{-1}-(4x)(x^2+3)^{-2}

With f'(x)= (4x)(x^2+3)^{-1}- 1 the second derivative is
4(x^2+ 3)^{-1}- (4x)(x^2+ 3)^{-2}(2x)= 4(x^2+ 3)^{-1}- 8x^2(x^2+3)^{-2}
= \frac{4(x^2+ 3)}{(x^2+ 3)^{-2}}- \frac{8x^2}{(x^2+ 3)^{-2}}= \frac{-8x^2+ 3}{(x^2+ 3)^{-2}}
The second derivative always exists but there are two places where it is 0.

 

[Swerting]

Thank you very much, I see it now.
I figured it was something in the calculation of the first to second, I just couldn't see it.