6. Given the function $f(x)=\frac{2 x^{2}+b x+c}{x^{2}+1}(b<0)$ with the range $[1,3]$.
( [ ) Find the values of the real numbers $b, c$;
( II ) Determine the monotonicity of the function $F(x)=\lg f(x)$ on $x \in[-1,1]$, and provide a proof;
( III ) If $t \in \mathbf{R}$, prove that:
$\left.\lg \frac{7}{5} \leqslant F\left(\left|t-\frac{1}{6}\right| \cdots\left|t+\frac{1}{6}\right|\right) \leqslant \lg \frac{13}{5}\right)$.
$((1) b=-2, c=2 ;(I I) F(x)$ is a decreasing function for $x \in[-1,1]$ 5.)