Question

Cho x>=0, y>=0

Chứng minh \(x+2y-2\sqrt{xy}+2\sqrt{x}-6\sqrt{y}+10>0\)

Answer 1

\(x-2\sqrt{x}\left(\sqrt{y}-1\right)+\left(\sqrt{y}-1\right)^2+2y-6\sqrt{y}+10-\left(y-2\sqrt{y}+1\right)>0\)

\(\left(\sqrt{x}-\sqrt{y}+1\right)^2+\left(y-4\sqrt{y}+4\right)+5>0\)

\(\left(\sqrt{x}-\sqrt{y}+1\right)^2+\left(\sqrt{y}-2\right)^2+5>0\) ( dpcm)