Question

CMR : \(a^4+b^4\ge\dfrac{1}{8}\left(a+b\right)^4\)

Answer 1

\(8(a^4+b^4)\ge\left(a+b\right)^4\)

\(\Leftrightarrow\)\(8a^4+8b^4\ge a^4+4a^3b+6a^2b^2+4ab^3+b^4\)

\(\Leftrightarrow\) \(7a^4+7b^4\ge4a^3b+6a^2b^2+4ab^3\)

\(\Leftrightarrow\)\(4a^3\left(a-b\right)+4b^3\left(b-a\right)+3\left(a^4-2a^2b^2+b^4\right)\ge0\)

\(\Leftrightarrow\) \(4\left(a^3-b^3\right)\left(a-b\right)+3\left(a^2-b^2\right)^2\ge0\)

\(\Leftrightarrow\) \(4\left(a-b\right)^2\left(a^2+ab+b^2\right)\ge0\) với mọi a,b

\(\Rightarrowđpcm\)