6. Find the minimum value of $f(x)=\sqrt{10 x^{2}+28 x+26}+\sqrt{10 x^{2}-18 x+13}$.
6. The original function can be transformed into
$f(x)=\sqrt{10}\left[\sqrt{\left(x+\frac{7}{5}\right)^{2}+\frac{16}{25}}+\sqrt{\left(x-\frac{9}{10}\right)^{2}+\frac{49}{100}}\right]$. Let $a=\left(x+\frac{7}{5}, \frac{4}{5}\right), b=\left(x-\frac{9}{10},-\frac{7}{10}\right)$,
then $f(x)=(|a|+|b|) \cdot \sqrt{10}$,
$$
a-b=\left(\frac{23}{10}, \frac{15}{10}\right),|a-b|=\frac{\sqrt{754}}{10} .
$$

Since $|a-b| \leqslant|a|+|b|$,
then $f(x) \geqslant|a-b| \cdot \sqrt{10}=\frac{\sqrt{1885}}{5}$,
when and only when $a$ and $b$ are in opposite directions, i.e., $a=\lambda b(\lambda<0)$.
$$
\text { By } \frac{x+\frac{7}{5}}{x-\frac{9}{10}}=\frac{\frac{4}{5}}{-\frac{7}{10}}=\lambda \text {, }
$$

we get $x=-\frac{13}{75}, \lambda=-\frac{8}{7}$.
Therefore, when $x=-\frac{13}{75}$, $f(x)_{\min }=\frac{\sqrt{1885}}{5}$.