6. Let $a, b, c$ be non-negative real numbers. Then
$$
\begin{array}{l} 
S= \sqrt{\frac{a b}{(b+c)(c+a)}}+ \\
\sqrt{\frac{b c}{(a+c)(b+a)}}+\sqrt{\frac{c a}{(b+c)(b+a)}}
\end{array}
$$

The minimum value of $S$ is $\qquad$, and the maximum value of $S$ is $\qquad$
6. $1, \frac{3}{2}$.
$$
\text { Let } \frac{a}{b+c}=u^{2}, \frac{b}{c+a}=v^{2}, \frac{c}{a+b}=w^{2} \text {. }
$$

Assume $a+b+c=1$.
$$
\begin{array}{l}
\text { Then } a=\frac{u^{2}}{1+u^{2}}, b=\frac{v^{2}}{1+v^{2}}, c=\frac{w^{2}}{1+w^{2}} \\
\Rightarrow \frac{u^{2}}{1+u^{2}}+\frac{v^{2}}{1+v^{2}}+\frac{w^{2}}{1+w^{2}}=1 \\
\Rightarrow(w)^{2}+(v w)^{2}+(u u)^{2}+2(w)(v w)(w u)=1 .
\end{array}
$$

Let $u v=\cos C, v w=\cos A, w u=\cos B$, and
$$
\begin{array}{l}
\angle A+\angle B+\angle C=\pi \\
\angle A, \angle B, \angle C \in\left[0, \frac{\pi}{2}\right] .
\end{array}
$$

Thus $S=\cos A+\cos B+\cos C$
$$
\begin{array}{l}
=2 \cos \frac{A+B}{2} \cdot \cos \frac{A-B}{2}+\cos C \\
\quad \leqslant 2 \sin \frac{C}{2}+\cos C=-2\left(\sin \frac{C}{2}-\frac{1}{2}\right)^{2}+\frac{3}{2} \\
\quad \leqslant \frac{3}{2},
\end{array}
$$

with equality holding if and only if $a=b=c$.
Similarly,
$$
\begin{array}{l}
\cos A+\cos B+\cos C \\
=2 \sin \frac{C}{2} \cdot \cos \frac{A-B}{2}+\cos C \\
=1+4 \sin \frac{C}{2}\left(\sin \frac{A}{2} \cdot \sin \frac{B}{2}\right) \geqslant 1 .
\end{array}
$$