Ta có:
\(\left(x+1\right)\left(y+1\right)\ge\left(\sqrt{xy}+1\right)^2\left(Bunhiacopxki\right)\)
\(x+y\ge2\sqrt{xy}\left(Côsi\right)\)
\(2\sqrt{xy}+\dfrac{1}{\sqrt{xy}}\ge2\sqrt{2}\left(Côsi\right)\)
\(1=x^2+y^2\ge2xy\Rightarrow\sqrt{xy}\le\dfrac{1}{\sqrt{2}}\)
\(\Rightarrow M\ge\dfrac{2\sqrt{xy}\left(\sqrt{xy}+1\right)^2}{xy}\)
\(=2\left(\sqrt{xy}+2+\dfrac{1}{\sqrt{xy}}\right)\)
\(=2\left(2\sqrt{xy}+\dfrac{1}{\sqrt{xy}}+2-\sqrt{xy}\right)\)
\(\ge2\left(2\sqrt{2}+2-\dfrac{1}{\sqrt{2}}\right)=4+3\sqrt{2}\)
\(MinM=4+3\sqrt{2}\Leftrightarrow x=y=\dfrac{1}{\sqrt{2}}\)
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