\(\text{Xét hiệu:}\)
\(\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}=\frac{y.\left(x+y\right)}{xy.\left(x+y\right)}+\frac{x.\left(x+y\right)}{xy.\left(x+y\right)}-\frac{4xy}{xy.\left(x+y\right)}\)
\(=\frac{y^2+xy}{x^2y+xy^2}+\frac{x^2+xy}{x^2y+xy^2}-\frac{4xy}{x^2y+xy^2}\)
\(=\frac{x^2-2xy+y^2}{x^2y+xy^2}=\frac{\left(x-y\right)^2}{x^2y+xy^2}\)
\(\text{Vì }\left(x-y\right)^2\ge0\text{ với mọi x;y và }x>0;y>0\)
\(\text{nên: }\frac{\left(x-y\right)^2}{x^2y+xy^2}\ge0\text{ với mọi x;y hay }\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}\ge0\text{ với mọi x;y}\)
\(\Leftrightarrow\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\text{ với mọi x;y}\)
\(\text{Xét hiệu:}\)
\(\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}=\frac{y.\left(x+y\right)}{xy.\left(x+y\right)}+\frac{x.\left(x+y\right)}{xy.\left(x+y\right)}-\frac{4xy}{xy.\left(x+y\right)}\)
\(=\frac{xy+y^2}{xy.\left(x+y\right)}+\frac{x^2+xy}{xy.\left(x+y\right)}-\frac{4xy}{xy.\left(x+y\right)}=\frac{x^2-2xy+y^2}{xy.\left(x+y\right)}=\frac{\left(x-y\right)^2}{xy.\left(x+y\right)}\)
\(\text{Vì }\left(x-y\right)^2\ge0\text{ với mọi x;y };x>0;y>0\)
\(\text{Nên }\frac{\left(x-y\right)^2}{xy.\left(x+y\right)}\ge0\text{ với mọi x;y}\)
\(\text{hay }\frac{1}{x}+\frac{1}{y}-\frac{4}{x+y}\ge0\text{ với mọi x;y }\Leftrightarrow\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\text{ với mọi x;y}\)
\(\text{Dấu "=" xảy ra khi x=y}\)
