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Áp dụng Cauchy
[tex]x^3+\frac{1}{27}+\frac{1}{27}\geq 3\sqrt[3]{x^3.\frac{1}{27}.\frac{1}{27}}=\frac{x}{3}[/tex] (1)
[tex]y^3+\frac{1}{27}+\frac{1}{27}\geq 3\sqrt[3]{y^3.\frac{1}{27}.\frac{1}{27}}=\frac{y}{3}[/tex] (2)
[tex]z^3+\frac{1}{27}+\frac{1}{27}\geq 3\sqrt[3]{z^3.\frac{1}{27}.\frac{1}{27}}=\frac{z}{3}[/tex] (3)
Từ (1) (2) (3) -> [tex]x^3+y^3+z^3+\frac{6}{27}\geq \frac{x+y+z}{3}=\frac{1}{3}\rightarrow x^3+y^3+z^3\geq \frac{1}{3}-\frac{6}{27}=\frac{1}{9}(dpcm)[/tex]