Example 8 Let $x \in \mathbf{R}$, find the minimum value of the function $f(x)=\sqrt{x^{2}-4 x+13}+$ $\sqrt{x^{2}-10 x+26}$.
$$
\begin{array}{l}
\text { Solve } f(x)=\sqrt{(x-2)^{2}+3}+\sqrt{(5-x)^{2}+1} . \\
\text { Let } \\
\text { Then } \\
\mid z_{1}=(x-2)+3 \mathrm{i}, z_{2}=(5-x)+\mathrm{i}, \\
\text { Then } \quad f(x)=\left|z_{1}\right|+\left|z_{2}\right| \geqslant\left|z_{1}+z_{2}\right|=\mid 3+4 \mathrm{i}, \\
\text { Therefore } \\
\min f(x)=5 .
\end{array}
$$

Let
$$
\begin{array}{l}
\text { When } \arg z_{1}=\arg z_{2}=\arg \left(z_{1}+z_{2}\right)=\arg (3+4 \mathrm{i})=\arg \tan \left(\frac{4}{3}\right) \\
\text { That is, when } x=\frac{17}{4} \text {, } \min f(x)=5 .
\end{array}
$$