1. The maximum value of the function $y=5 \sqrt{x-1}+\sqrt{10-2 x}$ is
1. $6 \sqrt{3}$.

Analysis: The domain of the function is $[1,5]$, and $y>0$
$$
\begin{aligned}
y & =5 \sqrt{x-1}+\sqrt{2} \cdot \sqrt{5-x} \\
& \left.\leq \sqrt{5^{2}+(\sqrt{2}}\right)^{2} \times \sqrt{(\sqrt{x-1})^{2}+(\sqrt{5-x})^{2}}=\sqrt{27 \times 4}=6 \sqrt{3}
\end{aligned}
$$

The equality holds if and only if $\sqrt{2} \cdot \sqrt{x-1}=5 \sqrt{5-x}$, i.e., $x=\frac{127}{27}$. Therefore, the maximum value of the function is $6 \sqrt{3}$.