Áp dụng BĐT Cô-si:
\(\dfrac{x^2}{x+1}+\dfrac{x+1}{9}\ge2\sqrt{\dfrac{x^2\left(x+1\right)}{9\left(x+1\right)}}=\dfrac{2}{3}x\)
\(\dfrac{y^2}{y+1}+\dfrac{y+1}{9}\ge2\sqrt{\dfrac{y^2\left(y+1\right)}{9\left(y+1\right)}}=\dfrac{2}{3}y\)
Cộng vế:
\(\dfrac{x^2}{x+1}+\dfrac{y^2}{y+1}+\dfrac{x+y+2}{9}\ge\dfrac{2}{3}\left(x+y\right)\)
\(\Leftrightarrow P+\dfrac{1+2}{9}\ge\dfrac{2}{3}.1\)
\(\Rightarrow P\ge\dfrac{1}{3}\)
\(P_{min}=\dfrac{1}{3}\) khi \(x=y=\dfrac{1}{2}\)
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