ez toán lớp 8
a.Áp dụng Cauchy
[tex]a+b\geq 2\sqrt{ab}\\b+c\geq 2\sqrt{bc}\\c+a\geq 2\sqrt{ca}\\\rightarrow (a+b)(b+c)(c+a)\geq 2\sqrt{ab}.2\sqrt{bc}.2\sqrt{ca}=8abc[/tex]
b. Vẫn dùng Cauchy
[tex]a+b+c\geq 3\sqrt[3]{abc}\\a^2+b^2+c^2\geq 3\sqrt[3]{a^2b^2c^2}\\\rightarrow (a+b+c)(a^2+b^2+c^2)\geq 3\sqrt[3]{abc}.3\sqrt[3]{a^2b^2c^2}=9\sqrt[3]{a^3b^3c^3}=9abc[/tex]
c. Áp dụng Cauchy
[tex]\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq 3.\frac{1}{\sqrt[3]{\left ( a+1 \right )\left ( b+1 \right )\left ( c+1 \right )}}[/tex]
[tex]\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}\geq 3.\sqrt[3]{\frac{abc}{\left ( a+1 \right )\left ( b+1 \right )\left ( c+1 \right )}}[/tex]
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