Áp dụng bất đẳng thức Bunyakovsky:
\(NL^2=\left(\sqrt{4x+2\sqrt{x}+1}+\sqrt{4y+2\sqrt{y}+1}+\sqrt{4z+2\sqrt{z}+1}\right)^2\)
\(\le\left(1^2+1^2+1^2\right)\left(4x+2\sqrt{x}+1+4y+2\sqrt{y}+1+4z+2\sqrt{z}+1\right)\)
\(=3\left(4x+4y+4z\right)+3\left(2\sqrt{x}+2\sqrt{y}+2\sqrt{z}\right)+3\left(1+1+1\right)\)
\(=12\left(x+y+z\right)+6\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)+9\)
\(=153+6\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)\)
Mặt khác,theo Bunyakovsky: \(\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)^2\le3\left(x+y+z\right)=36\)
\(\Rightarrow\sqrt{x}+\sqrt{y}+\sqrt{z}\le6\)
\(\Rightarrow153+6\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)\le153+36=189\)
\(\Rightarrow NL\le\sqrt{189}\)
Dấu "=" xảy ra khi: \(x=y=z=4\)
