1.
\(T=1.\sqrt{2a+b}+1.\sqrt{2b+c}+1.\sqrt{2c+a}\)
\(T\le\frac{1}{2}\left(1+2a+b\right)+\frac{1}{2}\left(1+2b+c\right)+\frac{1}{2}\left(1+2c+a\right)\)
\(T\le\frac{1}{2}\left[3\left(a+b+c\right)+3\right]=3\)
\(T_{max}=3\) khi \(a=b=c=\frac{1}{3}\)
2.
\(\Leftrightarrow x+y+z=2\sqrt{x+1}+2\sqrt{y-3}+2\sqrt{z-1}\)
\(\Leftrightarrow\left(x+1-2\sqrt{x-1}+1\right)+\left(y-3-2\sqrt{y-3}+1\right)+\left(z-1-2\sqrt{z-1}+1\right)=0\)
\(\Leftrightarrow\left(\sqrt{x+1}-1\right)^2+\left(\sqrt{y-3}-1\right)^2+\left(\sqrt{z-1}-1\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x+1}=1\\\sqrt{y-3}=1\\\sqrt{z-1}=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=4\\z=2\end{matrix}\right.\)