Question

cho x,y,z > 0, tìm giá trị nhỏ nhất của biểu thức.

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

Answer 1

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

\(P+3=\frac{x+y+z}{y+z}+\frac{x+y+z}{x+z}+\frac{x+y+z}{x+y}\)

\(P+3=\left(x+y+z\right)\left(\frac{1}{y+z}+\frac{1}{x+z}+\frac{1}{x+y}\right)\)

\(2\left(P+3\right)=\left[\left(x+y\right)+\left(y+z\right)+\left(z+x\right)\right]\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)\ge9\)

\(\Rightarrow P+3\ge\frac{9}{2}\Leftrightarrow P\ge\frac{3}{2}\)

\("="\Leftrightarrow x=y=z\)