\(\dfrac{1}{2x+y+z}=\dfrac{1}{x+y+x+z}\le\dfrac{1}{4}.\left(\dfrac{1}{x+y}+\dfrac{1}{x+z}\right)\)
\(\le\dfrac{1}{4}.\dfrac{1}{4}.\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{x}+\dfrac{1}{z}\right)=\dfrac{1}{16}.\left(\dfrac{2}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
Tuong tu : \(\dfrac{1}{x+2y+z}\le\dfrac{1}{16}.\left(\dfrac{2}{y}+\dfrac{1}{z}+\dfrac{1}{x}\right)\)
\(\dfrac{1}{x+y+2z}\le\dfrac{1}{16}.\left(\dfrac{2}{z}+\dfrac{1}{y}+\dfrac{1}{x}\right)\)
=> \(VT\le\dfrac{1}{16}.\left(\dfrac{2}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{2}{y}+\dfrac{1}{z}+\dfrac{1}{x}+\dfrac{2}{z}+\dfrac{1}{y}+\dfrac{1}{x}\right)\)
= \(\dfrac{1}{16}.\left[4.\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\right]=1\left(dpcm\right)\)
Áp dụng bđt Cauchy-Schwarz:
\(\dfrac{1}{2x+y+z}\le\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
\(\dfrac{1}{x+2y+z}\le\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)
\(\dfrac{1}{x+y+2z}\le\dfrac{1}{16}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{z}\right)\)
Cộng theo vế suy ra đpcm. \("="\Leftrightarrow x=y=z=\dfrac{3}{4}\)