12. Find the minimum value of the expression $x^{4}-2 x y+y^{4}$.
Solution 1. We have,

$$
\begin{aligned}
x^{4}-2 x y+y^{4}=x^{4}-2 x^{2} y^{2}+ & y^{4}+2 x^{2} y^{2}-2 x y+\frac{1}{2}-\frac{1}{2}= \\
& =\left(x^{2}-y^{2}\right)^{2}+2\left(x y-\frac{1}{2}\right)^{2}-\frac{1}{2} \geqslant-\frac{1}{2} .
\end{aligned}
$$

Equality holds if $x^{2}=y^{2}$ and $x y=\frac{1}{2}$, for example, when $x=y=\frac{1}{\sqrt{2}}$.

Answer: $-\frac{1}{2}$.

Topic: "Transformation of Algebraic Expressions"