Question

cho các số thực dương tm x+y+z=3 CM x^3 + y^3 +z^3 +xyz lớn hơn hoặc bằng 4

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Answer 1

\(x^3+y^3+z^3-3xyz=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right).\\ \)
\(=3\left(x^2+y^2+z^2-xy-yz-zx\right)\)
\(abc\ge\left(a+b-c\right)\left(b+c-a\right)\left(c+a-b\right) \\ \)
\(abc\ge\left(3-2a\right)\left(3-2b\right)\left(3-2c\right)=12\left(ab+bc+ca\right)-8abc-18\left(a+b+c\right)+27\\ \)
\(4abc\ge\frac{4}{9}\left(12\left(ab+bc+ca\right)-27\right)=\frac{16}{3}\left(ab+bc+ca\right)-12\)
\(a^3+b^3+c^3+abc\ge3\left(a^2+b^2+c^2\right)+\frac{7}{3}\left(ab+bc+ca\right)-12 =\frac{11}{6}\left(a^2+b^2+c^2\right)-\frac{3}{2}\ge4\\ \)