1/ \(P=\frac{1}{x+y+x+z}+\frac{1}{x+y+y+z}+\frac{1}{x+z+y+z}\)
\(P\le\frac{1}{4}\left(\frac{1}{x+y}+\frac{1}{x+z}+\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{x+z}+\frac{1}{y+z}\right)\)
\(P\le\frac{1}{2}\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{x+z}\right)\le\frac{1}{8}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{y}+\frac{1}{z}+\frac{1}{x}+\frac{1}{z}\right)\)
\(P\le\frac{1}{4}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=1\)
Dấu "=" xảy ra khi \(x=y=z=\frac{3}{4}\)
2/ ĐKXĐ: ...
\(\Leftrightarrow4x^2-8x\sqrt{x+1}+3\left(x+1\right)\le0\)
\(\Leftrightarrow\left(2x-\sqrt{x+1}\right)\left(2x-3\sqrt{x+1}\right)\le0\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x\ge\sqrt{x+1}\\2x\le3\sqrt{x+1}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge0\\4x^2-x-1\ge0\\4x^2-9x-9\le0\end{matrix}\right.\) \(\Rightarrow\frac{-1+\sqrt{17}}{8}\le x\le3\)
\(\Rightarrow x=\left\{1;2;3\right\}\Rightarrow\sum x^2=1+4+9=14\)
