\(log_{\frac{1}{2}}x+log_{\frac{1}{2}}y\le log_{\frac{1}{2}}\left(x+y^2\right)\Leftrightarrow log_{\frac{1}{2}}\left(xy\right)\le log_{\frac{1}{2}}\left(x+y^2\right)\Leftrightarrow xy\ge x+y^2\)
\(\Leftrightarrow x\left(y-1\right)\ge y^2\Leftrightarrow\left\{{}\begin{matrix}y>1\\x\ge\frac{y^2}{y-1}\end{matrix}\right.\)
\(\Rightarrow P=x+3y\ge\frac{y^2}{y-1}+3y=4y+1+\frac{1}{y-1}=4\left(y-1\right)+\frac{1}{y-1}+5\)
\(\Rightarrow P\ge2\sqrt{4\left(y-1\right).\frac{1}{\left(y-1\right)}}+5=9\)
\(\Rightarrow P_{min}=9\) khi \(\left\{{}\begin{matrix}x=\frac{9}{2}\\y=\frac{3}{2}\end{matrix}\right.\)
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