4. Let the function be
$$
\begin{array}{l}
f(x)=\sqrt{10-6 \cos x}+\sqrt{\frac{17}{8}-\frac{3 \sqrt{2}}{2} \sin x}+ \\
\sqrt{19-2 \sqrt{2} \cos x-8 \sin x} \text {. } \\
\end{array}
$$

For all real numbers $x$, the minimum value of $f(x)$ is
$\qquad$ .
4. $\frac{21 \sqrt{2}}{4}-1$.

Rewrite $f(x)$ as
$$
\begin{array}{l}
\sqrt{(\cos x-3)^{2}+\sin ^{2} x}+ \\
\sqrt{\cos ^{2} x+\left(\sin x-\frac{3 \sqrt{2}}{4}\right)^{2}}+ \\
\sqrt{(\cos x-\sqrt{2})^{2}+(\sin x-4)^{2}} .
\end{array}
$$

Let $A(3,0)$, $B\left(0, \frac{3 \sqrt{2}}{4}\right)$, $C(\sqrt{2}, 4)$, and $P(\cos x, \sin x)$. Then we need to find the minimum value of $P A+P B+P C$. Since the distance from $O(0,0)$ to $A B$ is 1, the unit circle centered at $O$ is tangent to $A B$, and let the point of tangency be $D\left(\frac{1}{3}, \frac{2 \sqrt{2}}{3}\right)$. Also, $O C$ passes through point $D$, so the minimum value of $P A+P B+P C$ is
$$
\begin{array}{l}
D A+D B+D C=A B+D C \\
=\frac{9 \sqrt{2}}{4}+(3 \sqrt{2}-1)=\frac{21 \sqrt{2}}{4}-1 .
\end{array}
$$