Question

CMR: \(x^2+2y^2-2xy+2x-4y+3>0\) với mọi x,y ∈ R

Answer 1

\(x^2+2y^2-2xy+2x-4y+3\)

\(=x^2-2x\left(y-1\right)+\left(y-1\right)^2-\left(y-1\right)^2+2y^2-4y+3\)

\(=\left(x-y+1\right)^2-y^2+2y+1+2y^2-4y+3\)

\(=\left(x-y+1\right)^2+y^2-2y+4\)

\(=\left(x-y+1\right)^2+\left(y-1\right)^2+3>0\forall x;y\)