Example 3 Find the maximum value of the function $y=6 \sqrt{x-5}+8 \sqrt{6-x}$.
Analysis: Observing that $(x-5)+(6-x)=1$, we can construct two sets of numbers $\sqrt{x-5}, \sqrt{6-x}; 6,8$, and use the Cauchy-Schwarz inequality to solve.
$$y=6 \sqrt{x-5}+8 \sqrt{6-x} \leqslant(x-5+6-x)^{\frac{1}{2}}$$
- $\left(6^{2}+8^{2}\right)^{\frac{1}{2}}=10, \therefore$ the maximum value sought is 10.