Áp dụng Bất đẳng thức Cauchy :
\(\dfrac{1}{x^2+y^2}+\dfrac{x^2+y^2}{4}\ge1\)
\(\dfrac{1}{xy}+xy\ge2\)
Cộng vế theo vế, ta được:
\(\dfrac{1}{x^2+y^2}+\dfrac{1}{xy}+\dfrac{x^2+y^2}{4}+xy\ge3\)
\(\Leftrightarrow P+\dfrac{x^2+y^2+4xy}{4}\ge3\)
\(\Leftrightarrow P+\dfrac{\left(x+y\right)^2+2xy}{4}\ge3\)
\(\Leftrightarrow P+\dfrac{4+2xy}{4}\ge3\Leftrightarrow P\ge3-\dfrac{4-2xy}{4}\) (vì: \(x+y=2\Rightarrow\left(x+y\right)^2=4\) )
Mà: \(x^2+y^2\ge2xy\Rightarrow x^2+y^2+2xy\ge4xy\Rightarrow4\ge4xy\Rightarrow2\ge2xy\)
\(\Rightarrow P=3-\dfrac{4+2xy}{4}\ge3-\dfrac{4-2}{4}=\dfrac{3}{2}\)
Vậy \(MinP=\dfrac{3}{2}\) khi \(x+y=1\)
Áp dụng BĐT AM-GM ta có:
\(x+y=2\ge2\sqrt{xy}\Rightarrow4\ge4xy\Rightarrow xy\le1\)
Áp dụng BĐT Cauchy-Schwarz dạng Engel ta có:
\(P=\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}+\dfrac{1}{2xy}\)
\(\ge\dfrac{\left(1+1\right)^2}{x^2+y^2+2xy}+\dfrac{1}{2xy}=\dfrac{4}{\left(x+y\right)^2}+\dfrac{1}{2xy}\)
\(\ge\dfrac{4}{4}+\dfrac{1}{2}=1+\dfrac{1}{2}=\dfrac{3}{2}\left(x+y=2;xy\le1\right)\)
Đẳng thức xảy ra khi \(x=y=1\)
Áp dụng Bất đẳng thức Svac:
\(P=\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}+\dfrac{1}{2xy}\ge\dfrac{4}{\left(x+y\right)^2}+\dfrac{1}{2xy}\)
\(\Rightarrow P\ge1+\dfrac{1}{2xy}\ge1+\dfrac{1}{2\cdot\dfrac{\left(x+y\right)^2}{2}}=1+\dfrac{1}{2}=\dfrac{3}{2}\)
Vậy MinP=3/2 khi x=y=1
