Example 4. Under the constraint conditions: $\left\{\begin{array}{l}x+2 y-2 \geqslant 0, \\ 2 x+y-2 \geqslant 0\end{array}\right.$, find the minimum value of $\log _{\theta}\left(3^{x}+3^{y}\right)$ and the corresponding values of $\mathrm{x}$ and $\mathrm{y}$.
Solve the system of inequalities representing the plane region as the shaded part (including the boundary) in Figure 4. The coordinates of point A are
$$
\left(\frac{2}{3}, \frac{2}{3}\right) \text {. }
$$

Thus, we have $\log _{9}\left(3^{x}+3^{y}\right)$
$$
\geqslant \log _{0} 2 \cdot \sqrt{ } 3^{x} \cdot 3^{y}=\log _{0} 2+\frac{1}{4}(x+y)
$$
(equality holds only when $3^{x}=$
$3^{y}$, i.e., $x=y$
). Let
$x+y=k$, then
when the line $y=-x$
+ moves within the shaded region
with at least one
common point, the
range of the y-intercept $k$ is $k \geqslant \frac{4}{3}$ (when the line passes through point A, i.e., $\mathrm{x}=\mathrm{y}=\frac{2}{3}$, the equality holds). Therefore,
$$
\begin{array}{c}
\log _{8}\left(3^{x}+3^{y}\right) \geqslant \frac{1}{4}(x+y)+\log _{8} 2 \\
\geqslant \frac{1}{4} \times \frac{4}{3} \log _{\theta} 2 \cdot
\end{array}
$$

Thus, when $x=y=\frac{2}{3}$, the minimum value of $\log _{\theta}\left(3^{x}+3^{y}\right)$ is $\frac{1}{3}+\log _{9} 2$.
(Author's affiliation: Luqiao Middle School, Huangyan County, Zhejiang Province)