Holder đi
[tex](a^7+b^7+c^7)(a+b+c)(a+b+c)\geq (a^3+b^3+c^3)^3\\\Rightarrow a^7+b^7+c^7\geq \frac{(a^3+b^3+c^3)^3}{(a+b+c)^2}[/tex]
[tex]a^3+b^3+c^3\geq \frac{(a+b+c)^3}{9} \\\Rightarrow (a+b+c)^3\leq 9(a^3+b^3+c^3)\\\Rightarrow (a+b+c)^2\leq \sqrt[3]{81(a^3+b^3+c^3)^2}[/tex]
Vậy [tex]\frac{(a^3+b^3+c^3)^3}{(a+b+c)^2}\geq \frac{(a^3+b^3+c^3)^3}{\sqrt[3]{81(a^3+b^3+c^3)^2}}=\frac{\sqrt[3]{(a^3+b^3+c^3)^9}}{\sqrt[3]{81(a^3+b^3+c^3)^2}}=\sqrt[3]{\frac{(a^3+b^3+c^3)^7}{81}}\geq \sqrt[3]{\frac{(\sqrt{3(a^3b^3+b^3c^3+c^3a^3)})^7}{81}}\geq \sqrt[3]{\frac{\sqrt{3}^7}{\sqrt{3}^8}}=\sqrt[3]{\frac{1}{\sqrt{3}}}[/tex]
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
