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

Toán Lớp 9: A=√x|√x+1 Tìm gtnn B với B=A(x-1)

Tháng Hai 11, 2023 Toán học 2 Comments 0 views

Toán Lớp 9: A=√x|√x+1
Tìm gtnn B với B=A(x-1)

Comments ( 2 )

  1. huongtructram
    huongtructram
    11 Tháng Hai, 2023 at 5:07 chiều
    Reply
    $A = \dfrac{\sqrt x}{\sqrt x + 1}$
    $B = A(x-1) ⇔ B = \dfrac{\sqrt x}{\sqrt x + 1}.(x-1)$
    $B = \dfrac{\sqrt x(x-1)}{\sqrt x + 1}$
    $B = \dfrac{\sqrt x(\sqrt x – 1)(\sqrt x +  1)}{\sqrt x+  1}$
    $B = \sqrt x(\sqrt x – 1)$
    $B =  x – \sqrt x$
    $B =\bigg( x – 2.\sqrt x . \dfrac{1}{2} + \dfrac{1}{4}\bigg) – \dfrac{1}{4}$
    $B = \bigg(\sqrt x – \dfrac{1}{2}\bigg)^2 – \dfrac{1}{4}$ 
    Vì $\bigg(\sqrt x – \dfrac{1}{2}\bigg)^2 \geq 0 ∀ x$
    ⇒ $\bigg(\sqrt x – \dfrac{1}{2}\bigg)^2 – \dfrac{1}{4} \geq \dfrac{-1}{4}$ 
    hay $B \geq \dfrac{-1}{4} ∀ x$
    Dấu “=” xảy ra ⇔ $\sqrt x – \dfrac{1}{2} = 0 ⇔ x = \dfrac{1}{4}$

  2. daolanhoa
    daolanhoa
    11 Tháng Hai, 2023 at 5:07 chiều
    Reply
    Giải đáp:
    $\min B = -\dfrac14 \Leftrightarrow x =\dfrac14$
    Lời giải và giải thích chi tiết:
    $\quad A = \dfrac{\sqrt x}{\sqrt x +1}\qquad (x \geqslant 0)$
    $\to B = A(x-1)=\dfrac{\sqrt x(x-1)}{\sqrt x+1}$
    $\to B =\dfrac{\sqrt x\left(\sqrt x -1\right)\left(\sqrt x +1\right)}{\sqrt x +1}$
    $\to B = \sqrt x\left(\sqrt x -1\right)$
    $\to B = x – \sqrt x$
    $\to B = \left(x – 2\cdot \dfrac12\cdot \sqrt x +\dfrac14\right) -\dfrac14$
    $\to B =\left(\sqrt x -\dfrac12\right)^2 -\dfrac14$
    Ta có:
    $\quad \left(\sqrt x -\dfrac12\right)^2 \geqslant 0 \quad \forall x\geqslant 0$
    $\to \left(\sqrt x -\dfrac12\right)^2 -\dfrac14 \geqslant -\dfrac14$
    $\to B \geqslant -\dfrac14$
    Dấu $=$ xảy ra $\Leftrightarrow \sqrt x -\dfrac12 =0 \Leftrightarrow x =\dfrac14$
    Vậy $\min B = -\dfrac14 \Leftrightarrow x =\dfrac14$

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

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