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\(\left(x^3+y^3\right)\left(x+y\right)=xy\left(1-x\right)\left(1-y\right)\Leftrightarrow\left(\frac{x^2}{y}+\frac{y^2}{x}\right)\left(x+y\right)=\left(1-x\right)\left(1-y\right)\left(1\right)\)
Ta có : \(\left(\frac{x^2}{y}+\frac{y^2}{x}\right)\left(x+y\right)\ge4xy\)
và \(\left(1-x\right)\left(1-y\right)=1-\left(x+y\right)+xy\le1-2\sqrt{xy}+xy\)
\(\Rightarrow1-2\sqrt{xy}+xy\ge4xy\Leftrightarrow0\) <\(xy\le\frac{1}{9}\)
Dễ chứng minh : \(\frac{1}{1+x^2}+\frac{1}{1+y^2}\le\frac{1}{1+xy};\left(x,y\in\left(0;1\right)\right)\)
\(\frac{1}{\sqrt{1+x^2}}+\frac{1}{\sqrt{1+y^2}}\le\sqrt{2\left(\frac{1}{1+x^2}+\frac{1}{1+y^2}\right)}\le\sqrt{2\left(\frac{2}{1+xy}\right)}=\frac{2}{\sqrt{1+xy}}\)
\(3xy-\left(x^2+y^2\right)=xy-\left(x-y\right)^2\le xy\)
\(\Rightarrow P\le\frac{2}{\sqrt{1+xy}}+xy=\frac{2}{\sqrt{1+t}}+t\), \(\left(t=xy\right)\), (0<\(t\le\frac{1}{9}\)
Xét hàm số :
\(f\left(t\right)=\frac{2}{\sqrt{t+1}}+t\) , (0<\(t\le\frac{1}{9}\)
Ta có: \(4xy\le\left(x+y\right)^2\le1\)
\(\Leftrightarrow xy\le\dfrac{1}{4}\)
\(A=\left(1+\dfrac{1}{x^2}\right)\left(1+\dfrac{1}{y^2}\right)\)
\(=\left(1+\dfrac{1}{4x^2}+\dfrac{1}{4x^2}+\dfrac{1}{4x^2}+\dfrac{1}{4x^2}\right)\left(1+\dfrac{1}{4y^2}+\dfrac{1}{4y^2}+\dfrac{1}{4y^2}+\dfrac{1}{4y^2}\right)\)
\(\ge5\sqrt[5]{\dfrac{1}{4^4x^8}}.5\sqrt[5]{\dfrac{1}{4^4y^8}}\)
\(=25\sqrt[5]{\dfrac{1}{4^8}.\dfrac{1}{\left(xy\right)^8}}\ge25\sqrt[5]{\dfrac{1}{4^8}.\dfrac{1}{\left(\dfrac{1}{4}\right)^8}}=25\)