Question

cho x,y là các số thực cmr 

\(2\left(x^2+1\right)\left(y^2+1\right)\ge\left(x+1\right)\left(y+1\right)\left(xy+1\right)\)

Answer 1

\(2\left(x^2+1\right)=\left(1+1\right)\left(x^2+1\right)>=\left(x+1\right)^2\)(bđt bunhiacopxki)(1)

\(2\left(y^2+1\right)=\left(1+1\right)\left(y^2+1\right)>=\left(y+1\right)^2\)(bđt bunhiacopxki)(2)

\(\left(x^2+1\right)\left(y^2+1\right)>=\left(xy+1\right)^2\)(3)

từ (1) (2) và (3)\(\Rightarrow\left(2\left(x^2+1\right)\left(y^2+1\right)\right)^2>=\left(\left(x+1\right)\left(y+1\right)\left(xy+1\right)\right)^2\)

\(\Rightarrow2\left(x^2+1\right)\left(y^2+1\right)>=\left(x+1\right)\left(y+1\right)\left(xy+1\right)\)

dấu = xảy ra khi x=y=1