6. Find the minimum value of the function $f(x)=\sqrt{x^{2}-8 x+25}+\sqrt{x^{2}-4 x+13}$.
Solution. By completing the square under each of the radicals, we transform the function into the form

$$
f(x)=\sqrt{(x-4)^{2}+9}+\sqrt{(x-2)^{2}+9}
$$

For this expression, a geometric interpretation can be proposed. Consider each of the radicals as the distance from the point with coordinates $(x ; 0)$ to the point with coordinates $(4 ; 3)$ in the case of the first radical, and to the point with coordinates $(2 ;-3)$ in the case of the second radical. Thus, the value of the function is equal to the length of the broken line connecting the three points mentioned above. Since these points can be arranged on a straight line, the minimum value of the function will be the length of the segment connecting the points with coordinates $(4 ; 3)$ and $(2 ;-3)$.

Answer: $2 \sqrt{10}$.