1. Let non-negative real numbers $a, b, c$ satisfy $a+b+c=1$. Find the maximum and minimum values of $S=\sqrt{4 a+1}+\sqrt{4 b+1}+\sqrt{4 c+1}$.
Let $f(x)=\sqrt{4 x+1}(0 \leqslant x \leqslant 1)$. Then
$$
\begin{array}{l}
f^{\prime}(x)=\frac{2}{\sqrt{4 x+1}}>0, \\
f^{\prime \prime}(x)=-\frac{4}{(4 x+1)^{\frac{3}{2}}}<0 .
\end{array}
$$

Therefore, $f(x)$ is a convex function on $x \in [0,1]$.
Thus, $f(x) \leqslant \frac{2 \sqrt{21}}{7}\left(x-\frac{1}{3}\right)+\frac{\sqrt{21}}{3}$.
Hence, $S \leqslant 3 \times \frac{\sqrt{21}}{3}=\sqrt{21}$,
$$
S \geqslant f(0)+f(0)+f(1)=2+\sqrt{5} .
$$