Question

Cho x, y, z > 0 và x+ y + z = 1. Chứng minh: x + 2y + z \(\ge\)4(1-x)(1-y)(1-z)

Answer 1

nx \(4\left(1-x\right)\left(1-y\right)\left(1-z\right)=4\left(y+z\right)\left(1-y\right)\left(1-z\right)\) 

ap dung bdt \(\left(a+b\right)^2\ge4ab\) ta co \(4\left(y+z\right)\left(1-z\right)\left(1-y\right)\le\left(y+z+1-z\right)^2\left(1-y\right)=\left(y+1\right)^2\left(1-y\right)\) \(=\left(y+1\right)\left(y+1\right)\left(1-y\right)=\left(y+1\right)\left(1-y^2\right)\le y+1\) =\(y+x+y+z=x+2y+z\left(dpcm\right)\)