Question

[Nguyễn Trần Trúc Ly]

Cho a^3+b^3+c^3=0

CMR:a^3b^3+2b^3c^3+3c^3a^3≤0

@ Mashiro Shiina

Answer 1

[Mashiro Shiina]

đề: \(a^3b^3+2b^3c^3+3c^3a^3\le0\) đúng ko?

Answer 2

[Mashiro Shiina]

\(L=a^3b^3+2b^3c^3+3c^3a^3\)

\(L=a^3b^3+2b^3c^3+2c^3a^3+c^3a^3\)

\(L=a^3\left(b^3+c^3\right)+2c^3\left(a^3+b^3\right)\)

Vì \(a^3+b^3+c^3=0\) nên: \(\left\{{}\begin{matrix}b^3+c^3=-a^3\\a^3+b^3=-c^3\end{matrix}\right.\)

nên: \(L=-a^6-2c^6\le0\)

Dấu "=" khi \(a^3=b^3=c^3=0\Leftrightarrow a=b=c=0\)