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to find this we have to first find y''.

so I used the quotient rule twice to get this

y'' = (8x^5 - 16x^3 - 24x)/(x^2 + 1)^4

to find the inflection points we set y'' = 0 and solve for x.

I have a question on this

while solving y'' = (8x^5 - 16x^3 - 24x)/(x^2 + 1)^4 = 0

i come across this step

(8x)(x^2 + 1)(x^2 - 3) = 0

so that would mean

8x = 0 this mean x = 0

(x^2 + 1) = 0 this means x = sqrt(-1)

(x^2 - 3) = 0 this means x = +-sqrt(3)

but do we just ignore the x = sqrt(-1) and conclude that the inflection points are x = 0, x = sqrt(3) and x = -sqrt(3)

So once we get the inflection points we use the sign charts to find concave up and concave down.

intervals where graph is concave up: (-sqrt(3), 0) & (sqrt(3), infinity)

intervals where graph is concave down: (-infinity, -sqrt(3)) & (0, sqrt(3))

are these intervals i found correct?