Test $\mathbf{G}$ Given $a, b, c, d \in \mathbf{R}$, and satisfying $a+b+c+d=3, a^{2}+2 b^{2}+$ $3 c^{2}+6 d^{2}=5$, try to find the maximum and minimum values of $a$.
Solve by Cauchy inequality
$$
\left(2 b^{2}+3 c^{2}+6 d^{2}\right)\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}\right) \geqslant(b+c+d)^{2},
$$

with equality if and only if $2 b=3 c=6 d$. Then

that is
$$
\begin{array}{c}
5-a^{2} \geqslant(3-a)^{2} . \\
2 a^{2}-6 a+4 \leqslant 0 \Rightarrow 1 \leqslant a \leqslant 2 .
\end{array}
$$
(i) $a=1$, if and only if $\frac{b}{3}=\frac{c}{2}=\frac{d}{1}=\frac{3-1}{6}=\frac{1}{3}$, i.e., $b=1, c=\frac{2}{3}$, $d=\frac{1}{3}$, then $a$ has the minimum value 1.
(ii) $a=2$, if and only if $\frac{b}{3}=\frac{c}{2}=\frac{d}{1}=\frac{3-2}{6}=\frac{1}{6}$, i.e., $b=\frac{1}{2}, c=\frac{1}{3}, d=\frac{1}{6}$, then $a$ has the maximum value 2.