Test Question D: The sum of four positive numbers is 4, and the sum of their squares is 8. Determine the maximum value of the largest of these four numbers.

Restate it as: Let \( a \geqslant b \geqslant c \geqslant d > 0 \), and satisfy
\[
a+b+c+d=4, \quad a^{2}+b^{2}+c^{2}+d^{2}=8,
\]

Find \(\max a=\) ?
$$
\begin{aligned}
\max a & =\frac{u+\sqrt{(n-1)\left(n v-u^{2}\right)}}{n} \\
& =\frac{4+\sqrt{(4-1)(4 \times 8-16)}}{4} \\
& =1+\sqrt{3} .
\end{aligned}
$$

In the above problem, taking $n=4, u=4, v=8$, we have
$$
\begin{aligned}
\max a & =\frac{u+\sqrt{(n-1)\left(n v-u^{2}\right)}}{n} \\
& =\frac{4+\sqrt{(4-1)(4 \times 8-16)}}{4} \\
& =1+\sqrt{3} .
\end{aligned}
$$