Example 15 The maximum value of the function $f(x)=\sqrt{x^{4}-3 x^{2}-6 x+13}$ $-\sqrt{x^{4}-x^{2}+1}$ is $\qquad$
(1992, National High School Mathematics Competition)
Analysis: $f(x)=\sqrt{(x-3)^{2}+\left(x^{2}-2\right)^{2}}-$ $\sqrt{(x-0)^{2}+\left(x^{2}-1\right)^{2}}$ can be constructed as: the difference of distances from a moving point $\left(x, x^{2}\right)$ to two fixed points $(3,2)$ and $(0,1)$. Since the trajectory of the moving point $\left(x, x^{2}\right)$ is the parabola $y=x^{2}$, it is easy to see from the graph that the maximum value is $\sqrt{10}$.
It can also be constructed as
$$
\begin{array}{l}
\left|(x-3)+\left(x^{2}-2\right) \mathrm{i}\right|-\left|x+\left(x^{2}-1\right) \mathrm{i}\right| \\
=\left|(3-x)+\left(2-x^{2}\right) \mathrm{i}\right|-\left|x+\left(x^{2}-1\right) \mathrm{i}\right| \\
\leqslant\left|(3-x)+x+\left[\left(2-x^{2}\right)+\left(x^{2}-1\right)\right] \mathrm{i}\right| \\
=\sqrt{10} .
\end{array}
$$