1. Find the minimum value of the function $f(x)=\sqrt{x^{2}+2 x+2}+\sqrt{x^{2}-10 x+34}$.
1. Given $f(x)=\sqrt{(x-1)^{2}+(0-1)^{2}}+\sqrt{(x-5)^{2}+\left[0-(-3)^{2}\right]}$, this expression indicates that $f(x)$ is the sum of the distances from a point $(x, 0)$ on the $x$-axis to two fixed points $A(1,1)$ and $B(5,-3)$. From the graph (omitted), we know that the intersection point $P$ of line $AB$ with the $x$-axis is the point that minimizes $f(x)$. Therefore, $|AB|=4 \sqrt{2}$ is the minimum value of $f(x)$.