Tìm GTNN của
\(A=x-\sqrt{x}\)
\(B=x-\sqrt{x-2005}\)
Giúp mình vs ạ
K
Khách
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Những câu hỏi liên quan
26 tháng 7 2017
đkxđ là \(x\ne1;x>0\)
\(Q=\frac{\sqrt{x}\left(\left(\sqrt{x}\right)^3-1\right)}{x+\sqrt{x}+1}-\frac{\sqrt{x}\left(2\sqrt{x}+1\right)}{\sqrt{x}}+\frac{2\left(x-1\right)}{\sqrt{x}-1}\)
\(Q=\frac{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{x+\sqrt{x}+1}-2\sqrt{x}-1+\frac{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}-1}\)
\(Q=x-\sqrt{x}-2\sqrt{x}-1+2\sqrt{x}+2=x-\sqrt{x}+1\)
gtnn \(x-\sqrt{x}+1=x-\frac{1}{2}.2.\sqrt{x}+\frac{1}{4}+\frac{3}{4}=\left(\sqrt{x}-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)
gtnn 3/4
ý c bạn tự làm nha mk chịu
CM
1
28 tháng 10 2019
\(A^2=2\left(x^2+1\right)+2\sqrt{\left(x^2+1\right)^2-x^2}.\)
\(=2\left(x^2+1\right)+2\sqrt{x^4+x^2+1}\)
Vì \(x^2\ge0\)\(\Rightarrow A^2\ge2+2=4\)\(\Rightarrow A\ge2\)
Dấu "=" xảy ra khi x=0
27 tháng 11 2023
ĐKXĐ: x>=0
\(Q=\dfrac{x-8}{\sqrt{x}+1}=\dfrac{x-1-7}{\sqrt{x}+1}\)
\(=\sqrt{x}-1-\dfrac{7}{\sqrt{x}+1}\)
=\(\sqrt{x}+1-\dfrac{7}{\sqrt{x}+1}-2\)
=>\(Q>=2\cdot\sqrt{\left(\sqrt{x}+1\right)\cdot\dfrac{7}{\sqrt{x}+1}}-2=2\sqrt{7}-2\)
Dấu '=' xảy ra khi \(\left(\sqrt{x}+1\right)^2=7\)
=>\(\sqrt{x}+1=\sqrt{7}\)
=>\(\sqrt{x}=\sqrt{7}-1\)
=>\(x=8-2\sqrt{7}\)
KN
27 tháng 10 2019
a) \(A=\sqrt{x^2-2x+1}+\sqrt{x^2-6x+9}\)
\(=\sqrt{\left(x-1\right)^2}+\sqrt{\left(x-3\right)^2}\)
\(=\left|x-1\right|+\left|x-3\right|\ge\left|\left(x-1\right)+\left(3-x\right)\right|=2\)
Vậy\(A_{min}=2\Leftrightarrow\left(x-1\right)\left(3-x\right)\ge0\)
\(TH1:\hept{\begin{cases}x-1\ge0\\3-x\ge0\end{cases}}\Leftrightarrow1\le x\le3\)
\(TH1:\hept{\begin{cases}x-1\le0\\3-x\le0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\le1\\x\ge3\end{cases}}\left(L\right)\)
Vậy \(A_{min}=2\Leftrightarrow1\le x\le3\)
NV
Nguyễn Việt Lâm
Giáo viên
23 tháng 4 2022
ĐKXĐ: \(\dfrac{3}{2}\le x\le3\)
\(A=\sqrt{2x-3}+\sqrt{6-2x}+\left(2-\sqrt{2}\right)\sqrt{3-x}\)
\(A\ge\sqrt{2x-3+6-2x}+\left(2-\sqrt{2}\right)\sqrt{3-x}\ge\sqrt{3}\)
\(A_{min}=\sqrt{3}\) khi \(3-x=0\Rightarrow x=3\)
\(A=1.\sqrt{2x-3}+\sqrt{2}.\sqrt{6-2x}\le\sqrt{\left(1+2\right)\left(2x-3+6-2x\right)}=3\)
\(A_{max}=3\) khi \(2x-3=\dfrac{6-2x}{2}\Rightarrow x=2\)
29 tháng 7 2023
a: ĐKXĐ: x>0; x<>1
\(Q=\dfrac{x+\sqrt{x}+\sqrt{x}}{x-1}:\dfrac{2\left(\sqrt{x}+1\right)-2+x}{x\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{x-1}\cdot\dfrac{x\left(\sqrt{x}+1\right)}{2\sqrt{x}+x}\)
\(=\dfrac{x}{\sqrt{x}-1}\)
b: Q>2
=>Q-2>0
=>\(\dfrac{x-\sqrt{x}+1}{\sqrt{x}-1}>0\)
=>căn x-1>0
=>x>1
29 tháng 7 2023
a) ĐK: \(\left\{{}\begin{matrix}x\ge0\\x\ne1\end{matrix}\right.\)
\(Q=\left(\dfrac{\sqrt{x}}{\sqrt{x}-1}+\dfrac{\sqrt{x}}{x-1}\right):\left(\dfrac{2}{x}-\dfrac{2-x}{x\sqrt{x}+x}\right)\)
\(=\left(\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\dfrac{\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\dfrac{2\left(\sqrt{x}+1\right)}{x\left(\sqrt{x}+1\right)}-\dfrac{2-x}{x\left(\sqrt{x}+1\right)}\right)\)
\(=\dfrac{x+\sqrt{x}+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}:\dfrac{2\sqrt{x}+2-2+x}{x\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x+2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}.\dfrac{x\left(\sqrt{x}+1\right)}{x+2\sqrt{x}}\)
\(=\dfrac{x}{\sqrt{x}-1}\)
b) Q>2 <=> \(\dfrac{x}{\sqrt{x}-1}>2\Leftrightarrow x>2\sqrt{x}-2\)
\(\Leftrightarrow x-2\sqrt{x}+2>0\)
\(\Leftrightarrow\left(\sqrt{x}-1\right)^2+1>0\)
\(\Leftrightarrow\left(\sqrt{x}-1\right)^2\ge0\Leftrightarrow\left[{}\begin{matrix}\sqrt{x}-1\le0\\\sqrt{x}-1\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\ge1\\x\le1\end{matrix}\right.\)
KL:.....
\(A=\left(x-\sqrt{x}+\frac{1}{4}\right)-\frac{1}{4}\)
\(A=\left(\sqrt{x}-\frac{1}{2}\right)^2-\frac{1}{4}\ge\frac{-1}{4}\)Dấu "=" xảy ra khi \(\sqrt{x}=\frac{1}{2}\Leftrightarrow x=\frac{1}{4}\)
\(B=\left(\left(x-2005\right)-\sqrt{x-2005}+\frac{1}{4}\right)+\frac{8019}{4}\)
\(B=\left(\sqrt{x=2005}-\frac{1}{2}\right)^2+\frac{8019}{4}\ge\frac{8019}{4}\)
Dấu "=" xảy ra khi \(\sqrt{x-2005}=\frac{1}{2}\Rightarrow x-2005=\frac{1}{4}\Leftrightarrow x=\frac{8021}{4}\)