1. Ta có: [tex]\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Leftrightarrow ab+bc+ca=0[/tex]
Lại có: [tex]\sqrt{a+b}=\sqrt{a+c}+\sqrt{b+c}\Leftrightarrow a+c=a+c+b+c+2\sqrt{(a+c)(b+c)}\Leftrightarrow 2c+2\sqrt{(a+c)(b+c)}=0\Leftrightarrow c+\sqrt{(a+c)(b+c)}=0\Leftrightarrow \sqrt{(a+c)(b+c)}=-c\Leftrightarrow(a+c)(b+c)=c^2\Leftrightarrow ab+bc+ca+c^2=c^2\Leftrightarrow ab+bc+ca=0[/tex]
Vậy ta có đpcm.
2. Ta có: [tex]xy+4\leq 2y\Leftrightarrow x+\frac{4}{y}\leq 2\Leftrightarrow 2\geq x+\frac{4}{y}\geq 2\sqrt{x.\frac{4}{y}}=4\sqrt{\frac{x}{y}}\Rightarrow \frac{x}{y}\leq \frac{1}{4}[/tex]
Lại thấy:[tex]P=\frac{xy}{x^2+2y^2}\Rightarrow \frac{1}{P}=\frac{x^2+2y^2}{xy}=\frac{x}{y}+ 2.\frac{y}{x} =\frac{x}{y}+ \frac{y}{16x}+\frac{31}{16}.\frac{y}{x}\geq 2\sqrt{\frac{x}{y}.\frac{y}{16x}}+\frac{31}{16}.4=\frac{1}{2}+\frac{31}{4}=\frac{33}{4}\Rightarrow P\leq \frac{4}{33}[/tex]
Dấu "=" xảy ra khi và chỉ khi [tex]\left\{\begin{matrix} x=\frac{4}{y}\\ \frac{x}{y}=\frac{1}{4}\\ xy+4=2y \end{matrix}\right.\Leftrightarrow x=1,y=4[/tex]