Cho $a\geq 2, b\geq 3, c\geq 4 . Tìm GTLN P=\frac{ab\sqrt{c-4}+bc\sqrt{a-2}+ac\sqrt{b-3}}{abc}$
Cho $a\geq 2, b\geq 3, c\geq 4 . Tìm GTLN P=\frac{ab\sqrt{c-4}+bc\sqrt{a-2}+ac\sqrt{b-3}}{abc}$
Started By thuhanhthuhang, 20-07-2014 - 19:32
bbđt và cực trị
#1
Posted 20-07-2014 - 19:32
#2
Posted 20-07-2014 - 19:43
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 $\left\{\begin{matrix} a=4\\ b=6 \\ c=8 \end{matrix}\right.$
- shinichikudo201, Phuong Mark, PolarBear154 and 2 others like this
Sống là cho, đâu chỉ nhận riêng mình
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