Example 4 Let the function be
$$
\begin{array}{l}
f(x)=\sqrt{10-6 \cos x}+\sqrt{\frac{17}{8}-\frac{3 \sqrt{2}}{2} \sin x}+ \\
\sqrt{\frac{17}{8}-\frac{\sqrt{2}}{2} \cos x-2 \sin x} \text {. } \\
\end{array}
$$

Then the minimum value of $f(x)$ is $\qquad$
Notice,
$$
\begin{aligned}
f(x)= & \sqrt{(\cos x-3)^{2}+\sin ^{2} x}+ \\
& \sqrt{\cos ^{2} x+\left(\sin x-\frac{3 \sqrt{2}}{4}\right)^{2}}+ \\
& \sqrt{\left(\cos x-\frac{\sqrt{2}}{4}\right)^{2}+(\sin x-1)^{2}} .
\end{aligned}
$$

As shown in Figure 4, let $A(3,0), B\left(0, \frac{3 \sqrt{2}}{4}\right), C\left(\frac{\sqrt{2}}{4}, 1\right)$, and $P(\cos \theta, \sin \theta)$.

Thus, $f(\theta)=|P A|+|P B|+|P C|$, where point $P$ lies on the circle $\odot O\left(x^{2}+y^{2}=1\right)$.

Since $\odot O$ is tangent to $A B$ at point $D\left(\frac{1}{3}, \frac{2 \sqrt{2}}{3}\right)$, and points $O, D, C$ are collinear, we have:
$$
\begin{array}{l}
|P C| \geqslant|D C|,|P A|+|P B| \geqslant|A B| . \\
\text { Then }|P A|+|P B|+|P C| \geqslant|A B|+|D C| \\
=\frac{9 \sqrt{2}}{4}+\left(\frac{3 \sqrt{2}}{4}-1\right)=3 \sqrt{2}-1,
\end{array}
$$

The equality holds if and only if point $P$ coincides with $D$.