Example 1. Let $A, B, C$ be real numbers. Try to find the sufficient and necessary conditions for the inequality $A(x-y)(x-z)+B(y-z)(y-x)$ $+C(z-x)(z-y) \geqslant 0$
to hold for any real numbers $x, y, z$.
Solve: Rewrite (1) as
$$
\begin{array}{l}
A(x-y)^{2}-(B-A-C)(x-y)(y-z) \\
+C(y-z)^{2} \geqslant 0 .
\end{array}
$$

Let $t=\frac{x-y}{y-z}(y \neq z)$ in the above equation, we get
$$
A t^{2}-(B-A-C) t+C \geqslant 0
$$

which holds for any real number $t$ if and only if
$$
\begin{array}{l}
\left\{\begin{array}{l}
A>0, \\
(B-A-C)^{2}-4 A C \leqslant 0
\end{array}\right. \\
\text { or }\left\{\begin{array}{l}
A=B-A-C=0, \\
C \geqslant 0
\end{array}\right. \\
\Leftrightarrow\left\{\begin{array}{l}
A>0, \\
A^{2}+B^{2}+C^{2} \leqslant 2(A B+B C+C A) ,
\end{array}\right. \\
\Leftrightarrow \text { or }\left\{\begin{array}{l}
B=C \geqslant 0, \\
A=0
\end{array}\right. \\
\Leftrightarrow\left\{\begin{array}{l}
A \geqslant 0, B \geqslant 0, C \geqslant 0, \\
A^{2}+B^{2}+C^{2} \leqslant 2(A B+B C+C A) .
\end{array}\right.
\end{array}
$$

Note: The inequality $a x^{4}+b x^{2}+c>0(\geqslant 0)$ holds for all $x \in R$ if and only if
$$
\left\{\begin{array}{l}
a>0, \\
b^{2}-4 a c \leqslant 0
\end{array}\right.
$$