Example 2. Given $\mathrm{x}+\mathrm{y}=1$, find the maximum value of $\sqrt{2 \mathrm{x}+1}$ $+\sqrt{2 y+1}$.
$$
\begin{array}{l}
\sqrt{2 x+1}+\sqrt{2 y+1} \\
\leqslant 2 \sqrt{\frac{(2 x+1)+(2 y+1)}{2}}=2 \sqrt{2}, \\
\text { i.e., the maximum value of } \sqrt{2 x+1}+\sqrt{2 y+1} \text { is } 2 \sqrt{2} \text { . } \\
\end{array}
$$