Question

[Lê Văn Hoàng]

cho x,y,z >0 và \(x+y+z\le2\).CMR

\(\frac{x}{2x+1}+\frac{y}{2y+1}+\frac{z}{2z+1}\le\frac{6}{7}\)

Answer 1

[Nguyễn Thiều Công Thành]

áp dụng bđt schwarts ta có:

\(\frac{1}{2x+1}+\frac{1}{2y+1}+\frac{1}{2z+1}\ge\frac{\left(1+1+1\right)^2}{2x+2y+2z+3}\ge\frac{9}{7}\)

\(\Rightarrow1-\frac{1}{2x+1}+1-\frac{1}{2y+1}+1-\frac{1}{2z+1}\le3-\frac{9}{7}\)

\(\Rightarrow\frac{2x}{2x+1}+\frac{2y}{2y+1}+\frac{2z}{2z+1}\le\frac{12}{7}\)

\(\Rightarrow\frac{x}{2x+1}+\frac{y}{2y+1}+\frac{z}{2z+1}\le\frac{6}{7}\left(Q.E.D\right)\)

dấu = xảy ra khi x=y=z=2/3