Question

CMR với mọi x,y ta có :

\(x^4+y^4\ge xy^3+x^3y\)

Answer 1

\(x^4+y^4\ge xy^3+x^3y\\ \Leftrightarrow x^4-xy^3+y^4-x^3y\ge0\\ \Leftrightarrow x^3\left(x-y\right)-y^3\left(x-y\right)\ge0\\ \Leftrightarrow\left(x-y\right)\left(x-y\right)\left(x^2+xy+y^2\right)\ge0\\ \Leftrightarrow\left(x-y\right)^2\left(x^2+xy+y^2\right)\ge0\\ \)

Ma \(\left(x-y\right)^2\ge0\left(\forall x,y\right)\\ x^2+xy+y^2>0\left(\forall x,y\right)\)\(\Rightarrow x^4+y^4\ge xy^3+x^3y\left(\forall x,y\right)\left(dpcm\right)\)