II. (16 points) Find the maximum and minimum values of the function $y=\frac{x-x^{3}}{\left(1+x^{2}\right)^{2}}$.

 untranslated part:
(16 分) 求函数 $y=\frac{x-x^{3}}{\left(1+x^{2}\right)^{2}}$ 的最大值和最小值.

Note: The untranslated part is kept as is, since it appears to be a direct repetition of the problem statement in Chinese. If you need the entire text translated, including the repeated part, please let me know.
3. $\left[0, \frac{\sqrt{3}}{3}\right]$.

Given $x \in[-1,1]$, let $x=\cos \alpha(\alpha \in[0, \pi])$.
Then $y=\frac{\sin \alpha}{2+\cos \alpha}$
$$
\begin{aligned}
\Rightarrow & y \geqslant 0, \\
& 2 y=\sin \alpha-y \cos \alpha \\
& =\sqrt{1+y^{2}} \sin (\alpha+\theta) \leqslant \sqrt{1+y^{2}} .
\end{aligned}
$$

Thus, $1+y^{2} \geqslant 4 y^{2} \Rightarrow y^{2} \leqslant \frac{1}{3}$.
Since $y \geqslant 0$, therefore, $y \in\left[0, \frac{\sqrt{3}}{3}\right]$.