Example 1 Let $a, b, c, d \geqslant 0$, satisfying $\sum a=32$. Find the maximum and minimum values of the function
$$
f(a, b, c, d)=\sum \sqrt{5 a+9}
$$
From Cauchy-Schwarz inequality, we have
$$
\begin{array}{l}
\sum \sqrt{5 a+9}=\sum(1 \cdot \sqrt{5 a+9}) \\
\leqslant \sqrt{\left(1^{2}+1^{2}+1^{2}+1^{2}\right) \sum(5 a+9)} \\
=\sqrt{4\left(5 \sum a+9 \times 4\right)}=28 \\
\Rightarrow f(a, b, c, d)_{\max }=28 .
\end{array}
$$

At this point, $a=b=c=d=8$.
From the equation in the problem, we know
$$
\begin{array}{l}
a, b, c, d \in[0,32] \\
\Rightarrow 25 x(x-32) \leqslant 0 \\
\Rightarrow \sqrt{5 x+9} \geqslant \frac{5}{16} x+3 \\
\Rightarrow f(a, b, c, d) \geqslant \sum\left(\frac{5}{16} a+3\right)=22 \\
\Rightarrow f(a, b, c, d)_{\min }=22 .
\end{array}
$$

At this point, $a=32, b=c=d=0$ and its cyclic permutations.

The maximum value of $f(a, b, c, d)$ is found using the Cauchy-Schwarz inequality, and the minimum value of $f(a, b, c, d)$ is found using the tangent line method.