Question

1) Chứng minh BĐT: \(\frac{a+b}{2}-\sqrt{ab}< \frac{\left(a-b\right)^2}{8b}\) với a>b>0.

2) Cho ba số dương x, y, z thỏa mãn \(\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge2.\) Tìm GTLN càu P=xyz.

Answer 1

2 )\(\frac{1}{1+x}\ge\left(1-\frac{1}{1+y}\right)+\left(1-\frac{1}{1+z}\right)=\frac{y}{1+y}+\frac{z}{1+z}\ge2\sqrt{\frac{yz}{\left(1+y\right)\left(1+z\right)}}\)

CMTT \(\frac{1}{1+y}\ge2\sqrt{\frac{xy}{\left(1+x\right)\left(1+y\right)}};\frac{1}{1+z}\ge2\sqrt{\frac{xy}{\left(1+x\right)\left(1+y\right)}}\)

Nhân vế với vế 3 bđt được

\(\frac{1}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\ge\frac{8xyz}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\)

\(\Rightarrow P=xyz\le\frac{1}{8}\)

Dấu "=" xảy ra khi z=y=z = 1/2

Answer 2

1)

\(\Leftrightarrow\frac{\left(a-b\right)^2}{8b}>\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{2}\Leftrightarrow\frac{a-b}{2\sqrt{b}}>\sqrt{a}-\sqrt{b}\)

\(\Leftrightarrow a-2\sqrt{ab}+b>0\Leftrightarrow\left(\sqrt{a}-\sqrt{b}\right)^2>0\) (có a>b>0 theo gt) (đpcm)