Question

2)Cho x,y,z>0 và x+y+z=1 CMR:x+2y+z lớn hơn hoặc bằng 4.(1-x).(1-y).(1-z)

Answer 1

Từ x+y+z=1 => 1-x = y+z

Áp dụng BĐT \(\left(a+b\right)^2\ge4ab\), ta có :  \(4\left(1-x\right)\left(1-y\right)\left(1-z\right)=4\left(y+z\right)\left(1-z\right)\left(1-y\right)\le\left[\left(y+z\right)+\left(1-z\right)\right]^2.\left(1-y\right)\)

\(\Rightarrow4\left(y+z\right)\left(1-y\right)\left(1-z\right)\le\left(1+y\right)^2\left(1-y\right)=\left(1+y\right)\left(1-y^2\right)\le1+y\)

\(\Rightarrow1+y=x+2y+z\ge4\left(1-x\right)\left(1-y\right)\left(1-z\right)\)(ĐPCM)