8 The function $f(x)=\frac{x-x^{3}}{1+2 x^{2}+x^{4}}$ has a product of its maximum and minimum values of

保留源文本的换行和格式，翻译结果如下：

8 The function $f(x)=\frac{x-x^{3}}{1+2 x^{2}+x^{4}}$ has a product of its maximum and minimum values of
$8-\frac{1}{16}$ Hint: Let $x=\tan \theta, \theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then
$$
\begin{aligned}
f(x) & =\frac{\tan \theta\left(1-\tan ^{2} \theta\right)}{\left(1+\tan ^{2} \theta\right)^{2}} \\
& =\sin \theta \cos \theta \cos 2 \theta \\
& =\frac{1}{4} \sin 4 \theta \in\left[-\frac{1}{4}, \frac{1}{4}\right] .
\end{aligned}
$$

Therefore, the product of the maximum and minimum values $-\frac{1}{4} \times \frac{1}{4}=-\frac{1}{16}$.