3. If real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$, find the maximum value of $\left(x^{2}-y z\right)\left(y^{2}-z x\right)\left(z^{2}-x y\right)$.
(2016, Korean Mathematical Olympiad)
When $x^{2}-y z$, $y^{2}-z x$, and $z^{2}-x y$ are all greater than 0, by the AM-GM inequality we get
$$
\left(x^{2}-y z\right)\left(y^{2}-z x\right)\left(z^{2}-x y\right) \leqslant \frac{1}{8} ;
$$

When $x^{2}-y z$, $y^{2}-z x$, and $z^{2}-x y$ are one positive and two negative, by the AM-GM inequality we get
$$
\left(x^{2}-y z\right)\left(z x-y^{2}\right)\left(x y-z^{2}\right) \leqslant \frac{1}{8} .
$$

When $x=0, y=\frac{\sqrt{2}}{2}, z=-\frac{\sqrt{2}}{2}$, the maximum value $\frac{1}{8}$ is achieved.