Câu 1:
Ta có:
\(\frac{1}{x^2}+\frac{x^2}{100}\ge\frac{2}{10}\left(1\right)\)
\(\frac{1}{y^2}+\frac{y^2}{100}\ge\frac{2}{10}\left(2\right)\)
Lấy (1) + (2) vế theo vế ta được
\(\frac{1}{x^2}+\frac{1}{y^2}+\frac{x^2+y^2}{100}\ge\frac{2}{10}+\frac{2}{10}\)
\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}\ge\frac{4}{10}-\frac{x^2+y^2}{100}=\frac{4}{10}-\frac{20}{100}=\frac{1}{5}\)
Dấu = xaey ra khi \(x^2=y^2=10\)hay \(x=y=\sqrt{10}\)
Bài 2/ Ta có:
\(\hept{\begin{cases}x+y+z=1\left(1\right)\\\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1\left(2\right)\end{cases}}\)
Lấy (1) . (2) vế theo vế ta được
\(\left(x+y+z\right)\left(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\right)=1\)
\(=\frac{x^2}{y+z}+x+\frac{y^2}{z+x}+y+\frac{z^2}{x+y}+z=1\)
\(=\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}+\left(x+y+z\right)=1\)
\(=\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}+1=1\)
\(=\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=0\)