\(x\ge1;y\ge4\)
- Với \(x=1;y=4\Rightarrow M=0\)
- Với \(x\ne1;y\ne4\)
\(M=\frac{\sqrt{x-1}}{x}+\frac{\sqrt{y-4}}{y}=\frac{\sqrt{x-1}}{x-1+1}+\frac{\sqrt{y-4}}{y-4+4}\)
\(M=\frac{1}{\sqrt{x-1}+\frac{1}{\sqrt{x-1}}}+\frac{1}{\sqrt{y-4}+\frac{4}{\sqrt{y-4}}}\le\frac{1}{2\sqrt{\sqrt{x-1}.\frac{1}{\sqrt{x-1}}}}+\frac{1}{2\sqrt{\frac{4\sqrt{y-4}}{\sqrt{y-4}}}}=\frac{1}{2}+\frac{1}{4}=\frac{3}{4}\)
\(\Rightarrow M_{max}=\frac{3}{4}\) khi \(\left\{{}\begin{matrix}x-1=1\\y-4=4\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=2\\y=8\end{matrix}\right.\)
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