Question

Cho \(a+b+c\le3\) .cm : \(a^4+b^4+c^4\ge a^3+b^3+c^3\)

Answer 1

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

\(\Leftrightarrow a^4+b^4+c^4-\left(a^3+b^3+c^3\right)\ge0\)

\(\Leftrightarrow a^4+b^4+c^4-\left(a^3+b^3+c^3\right)+3-\left(a+b+c\right)\ge0\)

\(\Leftrightarrow a^4-a^3-a+1+b^4-b^3-b+1+c^4-c^3-c+1\ge0\)

\(\Leftrightarrow\left(a-1\right)^2\left(a^2+a+1\right)+\left(b-1\right)^2\left(b^2+b+1\right)+\left(c-1\right)^2\left(c^2+c+1\right)\ge0\) đúng với mọi a, b, c.

\(\Rightarrowđpcm\)