Example 4. The three dimensions $\mathrm{x}, \mathrm{y}, \mathrm{z}$ of a rectangular prism sum to a constant $\mathrm{L}$. Find the maximum values of the volume $\mathrm{V}$ and the surface area $\mathrm{S}$.
This is to find the extremum of the objective functions
$$
\begin{array}{l}
V=x y z \\
\text { and } S=2(x y+y z+z x) \\
\end{array}
$$
under the constraint
$$
x+y+z=L
$$

Notice: (1), (2), and (3) are symmetric with respect to $x$, $y$, and $z$, and $V$ and $S$ clearly have no minimum value. Therefore, when $x=y=z=\frac{L}{3}$, we have
$$
\mathrm{V}_{\text {max }}=\frac{\mathrm{L}^{3}}{27}, \quad \mathrm{~S}_{\text {max }}=\frac{2}{9} \mathrm{~L}^{2} .
$$

From this, we obtain the proposition:
If the sum of three positive numbers is constant, then when these three positive numbers are equal, not only is their product maximized (which is already known from textbooks), but also the sum of their pairwise products is maximized.
There is a more general proposition:
If the sum of $\mathrm{n}$ positive numbers is constant, then when these $\mathrm{n}$ positive numbers are equal, not only is their product maximized (which is already known from textbooks), but also the sum of any $\mathrm{r}$ ( $1<\mathrm{r}<\mathrm{n})$ of them is maximized.