\(x-\sqrt{x+6}=\sqrt{y+6}-y\)
\(\Leftrightarrow x+y=\sqrt{x+6}+\sqrt{y+6}\)
\(\Rightarrow\left(x+y\right)^2=\left(\sqrt{x+6}+\sqrt{y+6}\right)^2\)
\(\le\left(1+1\right)\left(x+6+y+6\right)\) (áp dụng bđt Cauchy Shwarz)
\(=2\left(x+y\right)+24\)
\(\Rightarrow\left(x+y\right)^2-2\left(x+y\right)-24\le0\)
Đặt \(x+y=m\)
\(\Rightarrow m^2-2m-24\le0\)
\(\Leftrightarrow\left(m-6\right)\left(m+4\right)\le0\)
\(\Leftrightarrow-4\le m\le6\)
\(\Rightarrow Max_P=6\)
Dấu "=" xảy ra khi \(x=y=3\)
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