Question

Chứng minh bất đẳng thức : \(x^2+y^2-xy\) lớn hơn hoặc bằng \(x+y-1\)

Answer 1

\(x^2+y^2-xy\ge x+y-1\)

\(\Leftrightarrow2x^2+2y^2-2xy\ge2x+2y-2\)

\(\Leftrightarrow2x^2+2y^2-2xy-2x-2y+2\ge0\)

\(\Leftrightarrow\left(x^2-2x+1\right)+\left(y^2-2y+1\right)+\left(x^2-2xy+y^2\right)\ge0\)

\(\Leftrightarrow\left(x-1\right)^2+\left(y-1\right)^2+\left(x-y\right)^2\ge0\)

Bat ddang thuc cuoiđung,cac phep biendddooii tren la tuong dduong nen BĐT cuoi ddung =>đpcm

xay ra--ddang--thuc khi x=y=1