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[Amasa]

Tìm GTLN của $P=\frac{yz\sqrt{x-1}+xz\sqrt{y-2}+xy\sqrt{z-3}}{xyz}$ với $x\geq 1,y\geq 2,z\geq 3$

[Hoang Dinh Nhat]

Ta có: $P=\frac{\sqrt{x-1}}{x}+\frac{\sqrt{y-2}}{y}+\frac{\sqrt{z-3}}{z}$

Có các đánh giá: $\frac{\sqrt{x-1}}{x}\leq \frac{1}{2}\Leftrightarrow (x-2)^2\geq 0$ 

$\frac{\sqrt{y-2}}{y}\leq \frac{\sqrt{2}}{4}\Leftrightarrow (y-4)^2\geq 0$

$\frac{\sqrt{z-3}}{z}\leq \frac{\sqrt{3}}{6}\Leftrightarrow (z-6)^2\geq 0$

$\Rightarrow P\leq \frac{1}{2}+\frac{\sqrt{2}}{4}+\frac{\sqrt{3}}{6}=\frac{6+3\sqrt{2}+2\sqrt{3}}{12}$

Đạt tại: $(x;y;z)=(2;4;6)$

Edited by Hoang Dinh Nhat, 01-08-2017 - 11:18.