Question

[Thần Đồng]

CMR

a, \(2\left(a^4+b^4\right)\ge\left(a+b\right)\left(a^3+b^3\right)\)

b, \(3\left(a^4+b^4+c^4\right)\ge\left(a+b+c\right)\left(a^3+b^3+c^3\right)\)

Answer 1

[Khôi Bùi]

a ) CM : \(a^4+b^4\ge a^3b+b^3a\)

Giả sử điều cần c/m là đúng

\(\Rightarrow a^4+b^4-a^3b-b^3a\ge0\)

\(\Rightarrow a^3\left(a-b\right)-b^3\left(a-b\right)\ge0\)

\(\Rightarrow\left(a^3-b^3\right)\left(a-b\right)\ge0\)

\(\Rightarrow\left(a-b\right)^2\left(a^2+ab+b^2\right)\ge0\)

Ta có : \(\left\{{}\begin{matrix}\left(a-b\right)^2\ge0\\a^2+ab+b^2=\left(a+\dfrac{b}{2}\right)^2+\dfrac{3b^2}{4}\ge0\end{matrix}\right.\)

\(\Rightarrow\left(a-b\right)^2\left(a^2+ab+b^2\right)\ge0\)

\(\Rightarrow a^4+b^4-a^3b-b^3a\ge0\)

\(\Rightarrow a^4+b^4\ge a^3b+b^3a\)

\(\Rightarrow2\left(a^4+b^4\right)\ge a^4+a^3b+b^4+b^3a\)

\(\Rightarrow2\left(a^4+b^4\right)\ge\left(a+b\right)\left(a^3+b^3\right)\)

\(\left(đpcm\right)\)

Answer 2

[Khôi Bùi]

b ) \(\left(a+b+c\right)\left(a^3+b^3+c^3\right)\)

\(=a^4+a^3b+a^3c+b^3a+b^4+b^3c+c^3a+c^3b+c^4\)

\(=\left(a^4+b^4+c^4\right)+\left(a^3b+b^3a\right)+\left(b^3c+c^3b\right)+\left(a^3c+c^3a\right)\)

CMTT như a ) : \(\left\{{}\begin{matrix}a^4+b^4\ge a^3b+b^3a\\b^4+c^4\ge b^3c+c^3b\\a^4+c^4\ge a^3c+c^3a\end{matrix}\right.\)

\(\Rightarrow2\left(a^4+b^4+c^4\right)\ge a^3b+b^3a+b^3c+c^3b+a^3c+c^3a\)

\(\Rightarrow3\left(a^4+b^4+c^4\right)\ge a^4+b^4+c^4+a^3b+b^3a+b^3c+c^3b+a^3c+c^3a\)

\(\Rightarrow3\left(a^4+b^4+c^4\right)\ge\left(a+b+c\right)\left(a^3+b^3+c^3\right)\left(đpcm\right)\)