+ ĐKXĐ : \(\left\{{}\begin{matrix}x\ge-3\\y\ge-4\end{matrix}\right.\)
\(gt\Rightarrow x+y=6\left(\sqrt{x+3}+\sqrt{4+y}\right)\le6\sqrt{2\left(x+y+7\right)}\)
\(\Rightarrow\left(x+y\right)^2\le72\left(x+y+7\right)\)
\(\Rightarrow\left(x+y\right)^2-72\left(x+y\right)-504\le0\)
\(\Rightarrow\left(x+y-36\right)^2\le1800\Rightarrow P\le36+30\sqrt{2}\)
max \(P=36+30\sqrt{2}\Leftrightarrow\left\{{}\begin{matrix}x+3=y+4\\x+y=36+30\sqrt{2}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\frac{37}{2}+15\sqrt{2}\\y=\frac{35}{2}+15\sqrt{2}\end{matrix}\right.\)
+ \(x+y=6\left(\sqrt{x+3}+\sqrt{y+4}\right)\)
\(\Rightarrow\left(x+y\right)^2=36\left(x+y+7+2\sqrt{\left(x+3\right)\left(y+4\right)}\right)\)
\(\Rightarrow\left(x+y\right)^2-36\left(x+y\right)-252=72\sqrt{\left(x+3\right)\left(y+4\right)}\ge0\)
\(\Rightarrow\left(x+y-42\right)\left(x+y+6\right)\ge0\Rightarrow x+y\ge42\)
Min \(P=42\Leftrightarrow\left\{{}\begin{matrix}\sqrt{\left(x+3\right)\left(y+4\right)}=0\\x+y=42\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=-3\\y=45\end{matrix}\right.\\\left\{{}\begin{matrix}x=46\\y=-4\end{matrix}\right.\end{matrix}\right.\)
