Do the same for this question as b^3 did for the previous.
Firstly check over your double derivative working out because I'm pretty sure you should have
 = \frac{-2x(x^2-3)}{(x^2+1)^3})
The whole term will be positive when both numerator and the denominator are positive or both the numerator and the denominator are negative.
<0)
for all
 \cup (\sqrt{3},\infty))
(Try sketching the graph, you can note that it is a negative cubic with x-intercepts at
)
^3)
is never less than zero. (You know that
for all
and that the cubed part will not change this)
Hence both are never less than zero at the same time.
We will now focus on when both are positive.
>0)
for all
 \cup (0,\sqrt{3}))
^3>0)
for all
Therefore the whole term is positive for all
 \cup (0,\sqrt{3}))
(the intersection of the two domains)
The whole term will be negative when either of the numerator and the denominator are negative and the other isn't.
As the denominator is never negative, we don't need to find the condition of numerator = positive and denominator = negative. Instead, just the condition of the numerator being negative and the denominator being positive.
As shown before these were:
for all
for all
Therefore the whole term is negative for all
(the intersection of the two domains)
Therefore we have:
>0)
for all
<0)
for all
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