\(\sqrt{x}+\sqrt{y}+\sqrt{z}=1\)
Tìm GTNN của :
\(T=\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}-\left(x-y\right)^2-\left(y-z\right)^2-\left(z-y\right)^2\)
\(VT-\frac{1}{3}\ge\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}-\left(x+y+z\right)-\left(x-y\right)^2-\left(y-z\right)^2-\left(x-z\right)^2\)
\(=\sum_{cyc}\left(\frac{x^2}{y}-2x+y-\left(x-y\right)^2\right)\)
\(=\sum_{cyc}\left(\left(x-y\right)^2\left(\frac{1-y}{y}\right)\right)\)
\(=\sum_{cyc}\left(\left(x-y\right)^2\left(\frac{\left(\sqrt{x}+\sqrt{z}\right)\left(\sqrt{x}+2\sqrt{y}+\sqrt{z}\right)}{y}\right)\right)\ge0\)
\(\rightarrow VT\ge\frac{1}{3}\)"=" <=> x=y=z=1/9
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