Question

Cmr: a(a-b)+b(b-c)+c(c-a)>=0 . Với mọi a.b.c

Answer 1

\(a\left(a-b\right)+b\left(b-c\right)+c\left(c-a\right)\ge0\)

\(\Leftrightarrow a^2-ab+b^2-bc+c^2-ac\ge0\)

\(\Leftrightarrow2a^2-2ab+2b^2-2bc+2c^2-2ac\ge0\)

\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ac+a^2\right)\ge0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\forall a,b,c\)

Đẳng thức xảy ra khi \(\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}}\)\(\Rightarrow\hept{\begin{cases}a=b\\b=c\\c=a\end{cases}\Rightarrow}a=b=c\)