ĐK:x$\geq$1;y$\geq$2;z$\geq$3Áp dụng BĐT Cauchy cho các số dương:x-1+1$\geq$2$\sqrt{x-1}$$\Rightarrow$$\frac{\sqrt{x-1}}{x}$$\leq$$\frac{1}{2}$(1)Tương tự:y-2+2$\geq$2$\sqrt{2}$.$\sqrt{y-2}$$\Rightarrow$$\frac{\sqrt{y-2}}{y}$$\leq$$\frac{1}{2\sqrt{2}}$(2)z-3+3$\geq$2$\sqrt{3}$.$\sqrt{z-3}$$\Rightarrow$$\frac{\sqrt{z-3}}{z}$$\leq$$\frac{1}{2\sqrt{3}}$(3)Từ(1)(2)(3)$\Rightarrow$A$\leq$$\frac{1}{2}$(1+$\frac{1}{\sqrt{2}}$+$\frac{1}{\sqrt{3}}$)Dấu''='' xra$\Leftrightarrow$x=2;y=4;z=6
ĐK:
$x
\geq
1;y
\geq
2;z
\geq
3$Áp dụng BĐT Cauchy cho các số dương:
$x-1+1
\geq
2\sqrt{x-1}\Rightarrow
\frac{\sqrt{x-1}}{x}
\leq
\frac{1}{2}$(1)Tương tự:
$y-2+2\geq
2
\sqrt{2}.\sqrt{y-2}\Rightarrow
\frac{\sqrt{y-2}}{y}
\leq
\frac{1}{2\sqrt{2}}$(2)
$z-3+3
\geq
2
\sqrt{3}.\sqrt{z-3}\Rightarrow
\frac{\sqrt{z-3}}{z}
\leq
\frac{1}{2\sqrt{3}}$(3)Từ
$(1)(2)(3)
\Rightarrow
A
\leq
\frac{1}{2}(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}})
$Dấu
$''=''
$ x
ảy ra
$\Leftrightarrow
x=2;y=4;z=6
.$