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222-9+11+12:2*14+14 = ? ( )

Toán Lớp 9: Cho `a,b,c>0` và `a+b+c=1` `Cm:sqrt((ab)/(ab+c))+sqrt((bc)/(bc+a))+sqrt((ac)/(ac+b))<=3/2`

Tháng Mười Một 11, 2022 Toán học 1 Comment 0 views

Toán Lớp 9: Cho a,b,c>0 và a+b+c=1
Cm:sqrt((ab)/(ab+c))+sqrt((bc)/(bc+a))+sqrt((ac)/(ac+b))<=3/2

Comments ( 1 )

  1. thaolanphuobg
    thaolanphuobg
    11 Tháng Mười Một, 2022 at 2:38 chiều
    Reply
    $VT=\sqrt{\dfrac{ab}{ab+c}}+\sqrt{\dfrac{bc}{bc+a}}+\sqrt{\dfrac{ac}{ac+b}}$
    $VT=\sqrt{\dfrac{ab}{ab+c\left( a+b+c \right)}}+\sqrt{\dfrac{bc}{bc+a\left( a+b+c \right)}}+\sqrt{\dfrac{ac}{ac+b\left( a+b+c \right)}}$
    $VT=\sqrt{\dfrac{ab}{\left( c+a \right)\left( c+b \right)}}+\sqrt{\dfrac{bc}{\left( a+b \right)\left( a+c \right)}}+\sqrt{\dfrac{ac}{\left( b+a \right)\left( b+c \right)}}$
    $VT\le \dfrac{1}{2}\left( \dfrac{a}{c+a}+\dfrac{b}{c+b} \right)+\dfrac{1}{2}\left( \dfrac{b}{a+b}+\dfrac{c}{a+c} \right)+\dfrac{1}{2}\left( \dfrac{a}{b+a}+\dfrac{c}{b+c} \right)$
    $VT\le \dfrac{1}{2}\left( \dfrac{a}{c+a}+\dfrac{c}{a+c} \right)+\dfrac{1}{2}\left( \dfrac{b}{a+b}+\dfrac{a}{b+a} \right)+\dfrac{1}{2}\left( \dfrac{b}{c+b}+\dfrac{c}{b+c} \right)$
    $VT\le \dfrac{1}{2}\cdot 1+\dfrac{1}{2}\cdot 1+\dfrac{1}{2}\cdot 1$
    $VT\le \dfrac{3}{2}$ (đpcm)
    Dấu “=” xảy ra khi $a=b=c=\dfrac{1}{3}$

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222-9+11+12:2*14+14 = ? ( )

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