Example 2 Given $a, b, c \in R^{+}$, and $a+b+c=1$, find the maximum value of $\sqrt{4 a+1}+\sqrt{4 b+1}+\sqrt{4 c+1}$.
Solution: By Cauchy-Schwarz inequality, we have ( $1 \cdot \sqrt{4 a+1}+1$.
$$\begin{array}{l}
\sqrt{4 b+1}+1 \cdot \sqrt{4 c+1})^{2} \leqslant\left(1^{2}+1^{2}+1^{2}\right)(4 a+1 \\
+4 b+1+4 c+1)=12(a+b+c)+9=21, \text { when and only when }
\end{array}$$

$a=b=c=\frac{1}{3}$, the equality holds, the maximum value sought is
$$\sqrt{21} .$$