\(4xy\le\left(x+y\right)^2\Rightarrow\left(x+y\right)^2\ge x+y+2\)
\(\Leftrightarrow\left(x+y\right)^2-\left(x+y\right)-2\ge0\)
\(\Leftrightarrow\left(x+y+1\right)\left(x+y-2\right)\ge0\)
\(\Leftrightarrow x+y-2\ge0\) (do \(x+y+1>0\))
\(\Leftrightarrow x+y\ge2\)
\(\Rightarrow P=\frac{3\left(x+y\right)}{4}+\frac{x+y}{4}+\frac{1}{x+y}\ge\frac{3}{4}.2+2\sqrt{\frac{x+y}{4\left(x+y\right)}}=\frac{5}{2}\)
\(\Rightarrow P_{min}=\frac{5}{2}\) khi \(x=y=1\)
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