\(A=\frac{\sqrt{x}+4}{\sqrt{x}-1}\) và \(B=\frac{3\sqrt{x}+1}{x+2\sqrt{x}-3}-\frac{2}{\sqrt{x}+3}\) với \(x\ge0;x\ne1\)
Tìm tất cả giá trị của x để \(\frac{A}{B}\ge\frac{x}{4}+5\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
Mới đc câu a ak, thog cảm nha, trih độ mih thấp lắm:
\(\frac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\frac{2b}{a-b}\)
=\(\frac{a+\sqrt{ab}-\sqrt{ab}+b}{a-b}-\frac{2b}{a-b}\)
=\(\frac{a+b-2b}{a-b}=\frac{a-b}{a-b}=1\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{\left(1+\sqrt{x}\right)^2-4\sqrt{x}}{\sqrt{x}-1}\) \(\left(x\ge0;x\ne1\right)\)
\(A=\frac{x+2\sqrt{x}+1-4\sqrt{x}}{\sqrt{x}-1}=\frac{x-2\sqrt{x}+1}{\sqrt{x}-1}=\frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}-1}=\sqrt{x}-1\)
và \(B=\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{2}}+\frac{2+\sqrt{2}}{\sqrt{x}+1}\)
\(B=\frac{\sqrt{3}\left(\sqrt{3}+2\right)}{\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{2}}+\frac{\sqrt{2}\left(\sqrt{2}+1\right)}{\sqrt{2}+1}\)
\(B=\sqrt{3}+2+\frac{1}{\sqrt{3}-\sqrt{2}}+\sqrt{2}\)
\(B=\sqrt{3}+\sqrt{2}+\frac{1}{\sqrt{3}-\sqrt{2}}+2\)
\(B=\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)+1}{\sqrt{3}-\sqrt{2}}+2\)
\(B=\frac{3-2+1}{\sqrt{3}-\sqrt{2}}+2\)
\(B=\frac{2}{\sqrt{3}-\sqrt{2}}+2\)
để A = B thì \(\sqrt{x}-1\)= \(\frac{2}{\sqrt{3}-\sqrt{2}}+2\)
\(\sqrt{x}=\frac{2}{\sqrt{3}-\sqrt{2}}+3\)
\(\sqrt{x}=\frac{2\left(\sqrt{3}+\sqrt{2}\right)}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}+3\)
\(\sqrt{x}=2\sqrt{3}+2\sqrt{2}+3\)
tới bước này tui bí :(( mong các bạn giỏi khác giúp bạn :D
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{\left(3\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}-\frac{14\sqrt{x}+4}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}-\frac{\left(2\sqrt{x}+3\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}\)
\(=\frac{3x+7\sqrt{x}-6}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}-\frac{14\sqrt{x}+4}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}-\frac{2x+5\sqrt{x}+3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}\)
\(=\frac{x-12\sqrt{x}-13}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}=\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-13\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}-13}{\sqrt{x}+3}\)
\(A=\frac{\sqrt{x}+3-16}{\sqrt{x}+3}=1-\frac{16}{\sqrt{x}+3}\ge1-\frac{16}{3}=-\frac{13}{3}\)
\(A_{min}=-\frac{13}{3}\) khi \(x=0\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Câu 1:
\(\frac{A}{B}\ge\frac{x}{4}+5\Leftrightarrow\frac{\sqrt{x}+4}{\sqrt{x}-1}:\frac{1}{\sqrt{x}-1}\ge\frac{x}{4}+5\)
\(\Rightarrow\sqrt{x}+4\ge\frac{x}{4}+5\Rightarrow x-4\sqrt{x}+4\le0\)
\(\Rightarrow\left(\sqrt{x}-2\right)^2\le0\Rightarrow\sqrt{x}-2=0\Rightarrow x=4\)
Câu 2:
Bạn coi lại đề, biểu thức B không hợp lý
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\left(\dfrac{3x-3\sqrt{x}-3}{x+\sqrt{x}-2}+\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}+2}\right)\) : \(\dfrac{1}{\sqrt{x}+2}\)
=\(\dfrac{3x-3\sqrt{x}-3+\sqrt{x}+2-\sqrt{x}+1}{(\sqrt{x}-1)(\sqrt{x}+2)}\) .\(\sqrt{x}+2\)
=\(\dfrac{(3x-3\sqrt{x})(\sqrt{x}+2)}{(\sqrt{x}-1)(\sqrt{x}+2)}\)
=\(\dfrac{3\sqrt{x}(\sqrt{x}-1)(\sqrt{x}+2)}{(\sqrt{x}-1)(\sqrt{x}+2)}\) =\(3\sqrt{x}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(B=\frac{3\sqrt{x}+1}{x+2\sqrt{x}-3}-\frac{2}{\sqrt{x}+3}\)
\(=\frac{3\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}-\frac{2\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)
\(=\frac{\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}=\frac{1}{\sqrt{x}-1}\)
b) \(\frac{A}{B}=\frac{\sqrt{x}+4}{\sqrt{x-1}}:\frac{1}{\sqrt{x}-1}=\sqrt{x}+4\)
Để \(\frac{A}{B}\ge\frac{x}{4}+5\)
\(\Leftrightarrow\sqrt{x}+4\ge\frac{x}{4}+5\)
\(\Leftrightarrow4\sqrt{x}+16\ge x+20\)
\(\Leftrightarrow x-4\sqrt{x}+4\le0\)
\(\Leftrightarrow\left(\sqrt{x}-2\right)^2\le0\)
Mà \(\left(\sqrt{x}-2\right)^2\ge0;\forall x\ge0\)
\(\Rightarrow\left(\sqrt{x}-2\right)^2=0\)
\(\Leftrightarrow x=4\)
Vậy ...
Ta có \(\frac{A}{B}=\frac{\sqrt{x}+4}{\sqrt{x}-1}:\left(\frac{3\sqrt{x}+1}{x+2\sqrt{x}-3}-\frac{2}{\sqrt{x}+3}\right)=\frac{\sqrt{x}+4}{\sqrt{x}-1}:\left[\frac{3\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}-\frac{2\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\right]=\frac{\sqrt{x}+4}{\sqrt{x}-1}:\frac{3\sqrt{x}+1-2\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}+4}{\sqrt{x}-1}:\frac{\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}+4}{\sqrt{x}-1}.\left(\sqrt{x}-1\right)=\sqrt{x}+4\)
Để \(\frac{A}{B}\ge\frac{x}{4}+5\) thì \(\sqrt{x}+4\ge\frac{x}{4}+5\Leftrightarrow\sqrt{x}\ge\frac{x}{4}+1\Leftrightarrow x-4\sqrt{x}+4\le0\Leftrightarrow\left(\sqrt{x}-2\right)^2\le0\)
Mà \(\left(\sqrt{x}-2\right)^2\ge0\)
Suy ra \(\sqrt{x}-2=0\Leftrightarrow\sqrt{x}=2\Leftrightarrow x=4\)(tm)
Vậy x=4 thì \(\frac{A}{B}\ge\frac{x}{4}+5\)
\(B=\frac{1}{\sqrt{x}-1}\) (tự rút gọn nha)
\(\frac{A}{B}\ge\frac{x}{4}+5\\ \sqrt{x}+4\ge\frac{x}{4}+5\\ \frac{x}{4}-\sqrt{x}+1\le0\\ x-4\sqrt{x}+4\le0\\ \left(\sqrt{x}-2\right)^2\le0\\ \Rightarrow\sqrt{x}-2=0\\ \Rightarrow x=4\)
Vậy để \(\frac{A}{B}\ge\frac{x}{4}+5\) thì x=4