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23 tháng 10 2021
a) ĐKXĐ: \(x>0\)
\(A=\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{x-\sqrt{x}+1}-\dfrac{\sqrt{x}\left(2\sqrt{x}+1\right)}{\sqrt{x}}+1\)
\(=x+\sqrt{x}-2\sqrt{x}-1+1=x-\sqrt{x}\)
\(A=x-\sqrt{x}=2\)
\(\Leftrightarrow\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)=0\)
\(\Leftrightarrow\sqrt{x}=2\Leftrightarrow x=4\left(tm\right)\)(do \(\sqrt{x}+1\ge1>0\))
b) \(A=x-\sqrt{x}=\sqrt{x}\left(\sqrt{x}-1\right)>0\)(do \(x>1\))
\(\Leftrightarrow A=x-\sqrt{x}=\left|A\right|\)
c) \(A=x-\sqrt{x}=\left(x-\sqrt{x}+\dfrac{1}{4}\right)-\dfrac{1}{4}\)
\(=\left(\sqrt{x}-\dfrac{1}{2}\right)^2-\dfrac{1}{4}\ge-\dfrac{1}{4}\)
\(minA=-\dfrac{1}{4}\Leftrightarrow\sqrt[]{x}=\dfrac{1}{2}\Leftrightarrow x=\dfrac{1}{4}\left(tm\right)\)
23 tháng 10 2021
\(a,A=\dfrac{x\left(x\sqrt{x}+1\right)}{x-\sqrt{x}+1}-\dfrac{\sqrt{x}\left(2\sqrt{x}+1\right)}{\sqrt{x}}+1\left(x>0\right)\\ A=\dfrac{x\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{x-\sqrt{x}+1}-2\sqrt{x}-1+1\\ A=x+\sqrt{x}-2\sqrt{x}=x-\sqrt{x}\\ A=2\Leftrightarrow x-\sqrt{x}-2=0\\ \Leftrightarrow\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)=0\\ \Leftrightarrow\sqrt{x}=2\left(\sqrt{x}>0\right)\\ \Leftrightarrow x=4\left(tm\right)\)
\(b,x>1\Leftrightarrow\sqrt{x}-1>0\\ \Leftrightarrow\left|A\right|=\left|x-\sqrt{x}\right|=\left|\sqrt{x}\left(\sqrt{x}-1\right)\right|=\sqrt{x}\left(\sqrt{x}-1\right)=A\left(\sqrt{x}>0\right)\)
\(c,A=x-\sqrt{x}+\dfrac{1}{4}-\dfrac{1}{4}=\left(\sqrt{x}-\dfrac{1}{2}\right)^2-\dfrac{1}{4}\ge-\dfrac{1}{4}\\ A_{min}=-\dfrac{1}{4}\Leftrightarrow\sqrt{x}=\dfrac{1}{2}\Leftrightarrow x=\dfrac{1}{4}\left(tm\right)\)
31 tháng 10 2021
1: ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\notin\left\{4;9\right\}\end{matrix}\right.\)
Ta có: \(A=\dfrac{2\sqrt{x}-9-x+9+2x-4\sqrt{x}+\sqrt{x}-2}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\)
31 tháng 10 2021
\(1,A=\dfrac{2\sqrt{x}-9-x+9+2x-3\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\\ A=\dfrac{x-\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\\ A=\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\left(x\ge0;x\ne4;x\ne9\right)\\ 2,A< 1\Leftrightarrow\dfrac{\sqrt{x}+1-\sqrt{x}+3}{\sqrt{x}-3}< 0\\ \Leftrightarrow\dfrac{4}{\sqrt{x}-3}< 0\Leftrightarrow\sqrt{x}-3< 0\Leftrightarrow0\le x< 9\)
9 tháng 12 2021
\(A^2=\left(\sqrt{1-x}+\sqrt{1+x}\right)^2\le\left(1^2+1^2\right)\left(1-x+1+x\right)=4\\ \Leftrightarrow A\le2\\ A_{max}=2\Leftrightarrow1-x=1+x\Leftrightarrow x=0\\ A^2=2+2\sqrt{1-x^2}\ge2\\ \Leftrightarrow A\ge\sqrt{2}\\ A_{min}=\sqrt{2}\Leftrightarrow1-x^2=0\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-1\end{matrix}\right.\)
Vậy \(\sqrt{2}\le A\le2\)
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
18 tháng 12 2022
A = \(\dfrac{2\sqrt{x}-1}{\sqrt{x}+1}\) ; Đk \(x\) \(\ge\) 0
A = 2 - \(\dfrac{3}{\sqrt{x}+1}\)
\(\sqrt{x}\) \(\ge\) 0 => \(\sqrt{x}\) + 1 \(\ge\) 1
\(\dfrac{3}{\sqrt{x}+1}\) \(\le\) \(\dfrac{3}{1}\)
\(\dfrac{3}{\sqrt{x}+1}\) \(\le\) 3
- \(\dfrac{3}{\sqrt{x}+1}\) \(\ge\) -3
2 - \(\dfrac{3}{\sqrt{x}+1}\) \(\ge\) 2 - 3
2 - \(\dfrac{3}{\sqrt{x}+1}\) \(\ge\) -1
A(min ) = -1 dấu bằng xảy ra khi \(\sqrt{x}\) = 0 hay \(x\) = 0
MP
1
28 tháng 7 2019
\(A=\sqrt{x-2\sqrt{x-1}}\)\(+5\sqrt{x+3-4\sqrt{x-1}}\)\(+8\sqrt{x+8-6\sqrt{x-1}}\)
\(=\sqrt{x-1-2\sqrt{x-1}+1}\)\(+5\sqrt{x-1-4\sqrt{x-1}+4}\)\(+8\sqrt{x-1-6\sqrt{x-1}+9}\)
\(=\sqrt{\left(\sqrt{x-1}-1\right)^2}\)\(+5\sqrt{\left(\sqrt{x-1}-2\right)^2}\)\(+8\sqrt{\left(\sqrt{x-1}-3\right)^2}\)
\(=\sqrt{x-1}-1+5\sqrt{x-1}-10+8\sqrt{x-1}-24\)
\(=16\sqrt{x-1}-35\)
\(A_{min}=-35\Leftrightarrow16\sqrt{x-1}=0\Rightarrow x=1\)
NV
Nguyễn Việt Lâm
Giáo viên
27 tháng 7 2021
ĐKXĐ: \(x\ge0;x\ne1\)
\(P=\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\dfrac{3\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{6\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)
b.
\(P=\dfrac{2\sqrt{x}-\sqrt{x}-1}{\sqrt{x}+1}=-1+\dfrac{2\sqrt{x}}{\sqrt{x}+1}\)
Do \(\left\{{}\begin{matrix}2\sqrt{x}\ge0\\\sqrt{x}+1>0\end{matrix}\right.\) \(\Rightarrow\dfrac{2\sqrt{x}}{\sqrt{x}+1}\ge0\)
\(\Rightarrow P\ge-1\)
\(P_{min}=-1\) khi \(x=0\)
27 tháng 7 2021
a) Ta có: \(P=\dfrac{\sqrt{x}}{\sqrt{x}-1}+\dfrac{3}{\sqrt{x}+1}-\dfrac{6\sqrt{x}-4}{x-1}\)
\(=\dfrac{x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)
12 tháng 1 2021
\(A=\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{\sqrt{x}-1}{\sqrt{x}+1}+\frac{3\sqrt{x}+1}{1-x}\)
a) ĐKXĐ : \(\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)
\(=\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{\sqrt{x}-1}{\sqrt{x}+1}-\frac{3\sqrt{x}+1}{x-1}\)
\(=\frac{\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{3\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{x+2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{3\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{x+2\sqrt{x}+1+x-2\sqrt{x}+1-3\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{2x-3\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{2x-2\sqrt{x}-\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{2\sqrt{x}\left(\sqrt{x}-1\right)-\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{\left(\sqrt{x}-1\right)\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{2\sqrt{x}-1}{\sqrt{x}+1}\)
b) Với x = 4 thỏa mãn ĐKXĐ
\(A=\frac{2\sqrt{4}-1}{\sqrt{4}+1}=\frac{4-1}{2+1}=\frac{3}{3}=1\)
c) Chưa nghĩ ra :<
26 tháng 7 2023
1:
a: \(A=\dfrac{\sqrt{x}+1-2}{\sqrt{x}+1}=1-\dfrac{2}{\sqrt{x}+1}\)
căn x+1>=1
=>2/căn x+1<=2
=>-2/căn x+1>=-2
=>A>=-2+1=-1
Dấu = xảy ra khi x=0
b: 
\(x-1\ge0\Rightarrow x\ge1\Rightarrow A=\sqrt{1+\sqrt{x-1}}\ge1\)dấu "=" khi : x=1 \(A_{min}=1\)