10. The minimum value of the function $z=\sqrt{2 x^{2}-2 x+1}+\sqrt{2 x^{2}-10 x+13}$ is
Answer $\sqrt{10}$.
Analysis Because
$$
z=\sqrt{2 x^{2}-2 x+1}+\sqrt{2 x^{2}-10 x+13}=\sqrt{(x-0)^{2}+(x-1)^{2}}+\sqrt{(x-2)^{2}+(x-3)^{2}}
$$

This is the sum of the distances from a point $(x, y)$ on the line $y=x$ to the point $(0,1)$ and to the point $(2,3)$. According to the principle of reflection, the minimum value of $z$ should be the distance from the point $(0,1)$ to the point $(2,3)$, which is $\sqrt{10}$.