Question

cho 3 số không âm x,y,z thỏa mãn 1/(1+2x)+1/(1+2y)+1/(1+2z) tìm max của xyz

Answer 1

Đề thiếu rồi b

Answer 2

Ta có:

\(\dfrac{1}{1+2x}=1-\dfrac{1}{1+2y}+1-\dfrac{1}{1+2z}=\dfrac{2y}{1+2y}+\dfrac{2z}{1+2z}\ge\dfrac{4\sqrt{yz}}{\sqrt{\left(1+2y\right)\left(1+2z\right)}}\left(@\right)\)

Tương tự ta có:

\(\left\{{}\begin{matrix}\dfrac{1}{1+2y}\ge\dfrac{4\sqrt{zx}}{\sqrt{\left(1+2z\right)\left(1+2x\right)}}\left(@A\right)\\\dfrac{1}{1+2z}\ge\dfrac{4\sqrt{xy}}{\sqrt{1+2x}.\sqrt{1+2y}}\left(@A@\right)\end{matrix}\right.\)

Từ (@); (@A); (@A@) ta suy ra

\(\dfrac{1}{\left(1+2x\right).\left(1+2y\right).\left(1+2z\right)}\ge\dfrac{64xyz}{\left(1+2x\right)\left(1+2y\right)\left(1+2z\right)}\)

\(\Rightarrow xyz\le\dfrac{1}{64}\)