Lời giải:

Đặt $xy=a; x+y=b$ thì ta có: \(\left\{\begin{matrix} b^2-2a=4\\ b^2\geq 4a\end{matrix}\right.\)

$A=\frac{xy}{x+y+2}=\frac{a}{b+2}=\frac{b^2-4}{2(b+2)}=\frac{b-2}{2}$
Từ $b^2\geq 4a$. Thay $4a=2(b^2-4)$ có:

$b^2\geq 2(b^2-4)$

$\Leftrightarrow b^2\leq 8\Rightarrow b\leq 2\sqrt{2}$

Do đó: $A=\frac{b-2}{2}\leq \frac{2\sqrt{2}-2}{2}=\sqrt{2}-1$

Vậy $A_{\max}=\sqrt{2}-1$

\(\left|xy\right|+\left|yz\right|+\left|zx\right|\)

Tìm Min nhầm :((

À Tìm Max đúng r :))

\(P=\frac{x\left(x+y+z\right)+yz}{y+z}+\frac{y\left(x+y+z\right)+zx}{z+x}+\frac{z\left(x+y+z\right)+xy}{x+y}\)

\(P=\frac{\left(x+y\right)\left(x+z\right)}{y+z}+\frac{\left(x+y\right)\left(y+z\right)}{z+x}+\frac{\left(x+z\right)\left(y+z\right)}{x+y}\)

\(P\ge\left(x+y\right)+\left(y+z\right)+\left(z+x\right)=2\left(x+y+z\right)=2\)

Dấu "=" xảy ra khi \(x=y=z=\frac{1}{3}\)

ra rồi ko cần nữa nha mn:))

Ta có: \(A=\frac{1}{x^2+y^2}+\frac{1}{xy}=\frac{1}{x^2+y^2}+\frac{1}{2xy}+\frac{1}{2xy}\)

\(\ge\frac{4}{x^2+2xy+y^2}+\frac{1}{\frac{\left(x+y\right)^2}{2}}=\frac{4}{\left(x+y\right)^2}+\frac{2}{\left(x+y\right)^2}\)

\(=\frac{6}{\left(x+y\right)^2}=6\)

Đẳng thức xảy ra khi \(x=y=\frac{1}{2}\)

Bài làm:

Ta có: \(x+y\ge2\sqrt{xy}\)(bất đẳng thức Cauchy)

\(\Leftrightarrow\sqrt{xy}\le\frac{x+y}{2}\)

\(\Leftrightarrow xy\le\frac{\left(x+y\right)^2}{4}=\frac{1}{4}\)

Áp dụng bất đẳng thức Cauchy Schwars ta được:

\(A=\frac{1}{x^2+y^2}+\frac{1}{xy}=\left(\frac{1}{x^2+y^2}+\frac{1}{2xy}\right)+\frac{1}{2xy}\)

\(\ge\frac{\left(1+1\right)^2}{x^2+2xy+y^2}+\frac{1}{2.\frac{1}{4}}=\frac{4}{\left(x+y\right)^2}+\frac{1}{\frac{1}{2}}\)

\(=\frac{4}{1^2}+2=6\)

Dấu "=" xảy ra khi: \(x=y=\frac{1}{2}\)

Theo bđt Svacxo thì :

\(A=\frac{1}{x^2+y^2}+\frac{2}{2xy}\ge\frac{\left(1+\sqrt{2}\right)^2}{1}=1+2\sqrt{2}+2\)

Dấu "=" xảy ra khi và chỉ khi \(x=y=0,5\)

Vậy \(A_{min}=1+2\sqrt{2}+2\)khi \(x=y=0,5\)

dấu = mình chọn bừa nha 

ĐKXĐ : \(x,y>0\)

a/ \(A=\left(\sqrt{x}+\frac{y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}-x}+\frac{x+y}{\sqrt{xy}}\right)\)

\(=\left(\frac{x+\sqrt{xy}+y-\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\right):\left(\frac{x\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right).\sqrt{x}}-\frac{y\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}.\sqrt{y}\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}-\frac{\left(x+y\right)\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\right)\)

\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\frac{x^2-x\sqrt{xy}-y\sqrt{xy}-y^2-x^2+y^2}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}=\frac{x+y}{\sqrt{x}+\sqrt{y}}:\frac{-\sqrt{xy}\left(x+y\right)}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\frac{x+y}{\sqrt{x}+\sqrt{y}}.\frac{-\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{x+y}=\sqrt{y}-\sqrt{x}\)

b/ Ta có ; \(4+2\sqrt{3}=\left(\sqrt{3}+1\right)^2\)

\(\Rightarrow B=\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{3}=\sqrt{3}+1-\sqrt{3}=1\)