Example 1 The maximum value of the function $y=3 \sqrt{x-1}+4 \sqrt{5-x}$ is $\square$ $\qquad$ .
According to the problem, $y>0$ and $x \in[1,5]$, so by the Cauchy-Schwarz inequality, we have: $\square$
$$y^{2}=(3 \sqrt{x-1}+4 \sqrt{5-x})^{2} \leqslant\left(3^{2}+\right.$$
$\left.4^{2}\right)\left[(\sqrt{x-1})^{2}+(\sqrt{5-x})\right]=100$, equality holds if and only if $\frac{\sqrt{x-1}}{3}=\frac{\sqrt{5-x}}{4}$, i.e., $x=\frac{61}{25}$. Therefore, the maximum value of the function $y=3 \sqrt{x-1}+4 \sqrt{5-x}$ is 10.