Chủ đề này có 1 trả lời

[thuhanhthuhang]

Cho $a\geq 2, b\geq 3, c\geq 4 . Tìm GTLN P=\frac{ab\sqrt{c-4}+bc\sqrt{a-2}+ac\sqrt{b-3}}{abc}$

[hoangmanhquan]

Cho $a\geq 2, b\geq 3, c\geq 4 . Tìm GTLN P=\frac{ab\sqrt{c-4}+bc\sqrt{a-2}+ac\sqrt{b-3}}{abc}$

Ta có:

$P=\frac{\sqrt{c-4}}{c}+\frac{\sqrt{a-2}}{a}+\frac{\sqrt{b-3}}{b}$

$=\frac{\sqrt{(c-4).4}}{2c}+\frac{\sqrt{(a-2).2}}{\sqrt{2}b}+\frac{\sqrt{(b-3).3}}{\sqrt{3}b}$

$\leq \frac{c}{4c}+\frac{a}{2\sqrt{2}a}+\frac{b}{2\sqrt{3}b}$

$=\frac{\sqrt{3}+\sqrt{6}+2}{4\sqrt{3}}$

Dấu $"="$ xảy ra $\left\{\begin{matrix} a=4\\ b=6 \\ c=8 \end{matrix}\right.$