cho 2 số dương $x$ va $y$ biết $x+y=1$ .Tìm GTNN của $B=(1-1/x^{2})(1-1/y^{2})$
hihi :D –  ๖ۣۜConan♥doyleღ 28-03-16 09:20 PM
má ơi! conandoyle là cha đẻ của sherlock homles là bảo giúp là sao! –  Confusion 28-03-16 09:10 PM
c yen oi lm ho em cai –  ๖ۣۜConan♥doyleღ 27-03-16 09:02 PM
Ly ơi đây là đáp án của mình, chắc là đúng hihi!!!
Ta có: $B=(1-\frac{1}{x^2})(1-\frac{1}{y^2})$
$B=\frac{(1-x^2)(1-y^2)}{x^2.y^2}$
Ta thấy: $x+y=1\Rightarrow x^2+y^2+2xy=1\Rightarrow \begin{cases}1-x^2=y^2+2xy=y(y+2x) \\ 1-y^2=x^2+2xy=x(x+2y) \end{cases}$
Do đó: $B=\frac{y(y+2x)x(x+2y)}{x^2.y^2}$
$B=(5xy+2x^2+2y^2)/xy$
$B=5+(2x^2+2y^2/xy)$
Mà $x^2+y^2\geq 2xy$ nên $B\geq 5+4=9$
Dấu ''='' cậu tự tìm hộ mình nha!!!
Có gì thắc mắc thì bảo mình, đúng thì tích giùm mình, hihi!!!
oh ra v keke –  ๖ۣۜConan♥doyleღ 27-03-16 10:11 PM
hihi!!!! –  ๖ۣۜLazer๖ۣۜD♥๖ۣۜGin 27-03-16 10:09 PM
ko p 2xy>=4, mà là x^2 y^2>=2xy nên (x^2 y^2)/xy>=2, do đó 2(x^2 y^2)/xy>=4 –  ๖ۣۜLazer๖ۣۜD♥๖ۣۜGin 27-03-16 10:09 PM
vi sao 2xy>=4 –  ๖ۣۜConan♥doyleღ 27-03-16 10:07 PM
ma toan cho mk hoi –  ๖ۣۜConan♥doyleღ 27-03-16 10:06 PM
kcj. Thi vio thôi mà!!!Thanks cậu :D!!! –  ๖ۣۜLazer๖ۣۜD♥๖ۣۜGin 27-03-16 10:03 PM
chuc mung nhe –  ๖ۣۜConan♥doyleღ 27-03-16 10:00 PM
ma thi HSG QG ak? –  ๖ۣۜConan♥doyleღ 27-03-16 09:59 PM
uk thanks nha –  ๖ۣۜConan♥doyleღ 27-03-16 09:57 PM
Theo BĐT Cauchy:$xy\leq \frac{(x+y)^{2}}{4}=\frac{1}{4}$
Ta có:B=$1-(\frac{1}{x^{2}}+\frac{1}{y^{2}})+\frac{1}{x^{2}y^{2}}$=$1-\frac{(x+y)^{2}-2xy}{x^{2}y^{2}}+\frac{1}{x^{2}y^{2}}$
$\Rightarrow$B=$1+\frac{2}{xy}\geq9$
Dấu''='' xra$\Leftrightarrow$$x=y=\frac{1}{2}$

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