Cho √x + √y + √z=1 Tim GNNN: x/√y + y/√z + z/√x

Question

Cho √x + √y + √z=1
Tim GNNN:
x/√y + y/√z + z/√x

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Elliana 1 năm 2021-07-05T17:39:37+00:00 1 Answers 7 views 0

Answers ( )

    0
    2021-07-05T17:41:33+00:00

    Giải thích các bước giải:

    Đặt $\sqrt{x}=a, \sqrt y=b,\sqrt z=c$

    $\to a+b+c=1$ 

    $\to P=\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}$

    $\to P+1=\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}+a+b+c$

    $\to P+1=(\dfrac{a^2}{b}+b)+(\dfrac{b^2}{c}+c)+(\dfrac{c^2}{a}+a)$

    $\to P+1\ge 2\sqrt{\dfrac{a^2}{b}.b}+2\sqrt{\dfrac{b^2}{c}.c}+2\sqrt{\dfrac{c^2}{a}.a}$

    $\to P+1\ge 2a+2b+2c=2\to P\ge 1$

    Dấu = xảy ra khi $a=b=c\to x=y=z=\dfrac 19$

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