cho biểu thức \(A=\left(\frac{\text{x+1}}{\text{x-1}}-\frac{\text{x-1}}{\text{x+1}}+\frac{\text{x^2-4x-1}}{\text{x^2-1}}\right)\div\frac{x}{\text{x+2019}}\)
A) tìm đkxd và rút gọn
B) với \(x\in Z\), tìm GTLN của A
giúp mik câu b gấp nhé các bn
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\(A=\left[\frac{x^2}{x^3-4x}+\frac{6}{6-3x}+\frac{1}{x+2}\right]:\left[x-2+\frac{10-x^2}{x+2}\right]\) ĐKXĐ : \(x\ne0;x\ne\pm2\)
\(A=\left[\frac{x^2}{x\left(x+2\right)\left(x-2\right)}-\frac{6}{3\left(x-2\right)}+\frac{1}{x+2}\right]:\left[\frac{x^2-4}{x+2}+\frac{10-x^2}{x+2}\right]\)
\(A=\left[\frac{3x^2}{3x\left(x+2\right)\left(x-2\right)}-\frac{6x\left(x+2\right)}{3x\left(x+2\right)\left(x-2\right)}+\frac{3x\left(x+2\right)}{3x\left(x+2\right)\left(x-2\right)}\right]:\frac{6}{x+2}\)
\(A=\left[\frac{3x^2-6x^2-12x+3x^2+6x}{3x\left(x+2\right)\left(x-2\right)}\right].\frac{x+2}{6}\)
\(A=\frac{-x}{3x\left(x-2\right)}\)
\(A=\frac{-1}{3x-6}\)
đk: x;y;z dương nhé
áp dụng bđt cosi ta có:
\(x^2+yz>=2\sqrt{x^2yz}=2x\sqrt{yz};y^2+xz>=2\sqrt{y^2xz}=2y\sqrt{xz};z^2+xy=2\sqrt{z^2xy}=2z\sqrt{xy}\)
\(\Rightarrow\frac{1}{x^2+yz}< =\frac{1}{2x\sqrt{yz}};\frac{1}{y^2+xz}< =\frac{1}{2y\sqrt{xz}};\frac{1}{z^2+xy}< =\frac{1}{2z\sqrt{xy}}\)
\(\Rightarrow\frac{1}{x^2+yz}+\frac{1}{y^2+xz}+\frac{1}{z^2+xy}< =\frac{1}{2x\sqrt{yz}}+\frac{1}{2y\sqrt{xz}}+\frac{1}{2z\sqrt{xy}}=\frac{1}{2}\left(\frac{1}{x\sqrt{yz}}+\frac{1}{y\sqrt{xz}}+\frac{1}{z\sqrt{xy}}\right)\left(1\right)\)
áp dụng bđt cosi ta có:
\(\frac{1}{xy}+\frac{1}{xz}>=2\cdot\sqrt{\frac{1}{xy}\cdot\frac{1}{xz}}=\frac{2}{x\sqrt{yz}};\frac{1}{xy}+\frac{1}{yz}>=2\cdot\sqrt{\frac{1}{xy}\cdot\frac{1}{yz}}=\frac{2}{y\sqrt{xz}};\)
\(\frac{1}{yz}+\frac{1}{xz}>=2\cdot\sqrt{\frac{1}{yz}\cdot\frac{1}{xz}}=\frac{2}{z\sqrt{xy}}\)
\(\Rightarrow\frac{1}{xy}+\frac{1}{xz}+\frac{1}{xy}+\frac{1}{yz}+\frac{1}{yz}+\frac{1}{xz}=\frac{2}{xy}+\frac{2}{yz}+\frac{2}{xz}>=\frac{2}{x\sqrt{yz}}+\frac{2}{y\sqrt{xz}}+\frac{2}{z\sqrt{xy}}\)
\(\Rightarrow\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}>=\frac{1}{x\sqrt{yz}}+\frac{1}{y\sqrt{xz}}+\frac{1}{z\sqrt{xy}}\)
\(\Rightarrow\frac{1}{2}\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}\right)>=\frac{1}{2}\left(\frac{1}{x\sqrt{yz}}+\frac{1}{y\sqrt{xz}}+\frac{1}{z\sqrt{xy}}\right)\left(2\right)\)
từ \(\left(1\right);\left(2\right)\Rightarrow\frac{1}{x^2+yz}+\frac{1}{y^2+xz}+\frac{1}{z^2+xy}>=\frac{1}{2}\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}\right)\left(đpcm\right)\)
dấu = xảy ra khi x=y=z
nhầm từ \(\left(1\right);\left(2\right)\Rightarrow\frac{1}{x^2+yz}+\frac{1}{y^2+xz}+\frac{1}{z^2+xy}< =\frac{1}{2}\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}\right)\)
ĐKXĐ : \(x\ne\pm1\)
a/ \(A=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}\right):\left(\frac{2}{x^2-1}-\frac{x}{x-1}+\frac{1}{x+1}\right)\)
\(=\frac{x^2+2x+1-\left(x^2-2x+1\right)}{\left(x-1\right)\left(x+1\right)}:\frac{2-x\left(x+1\right)+\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\)
\(=\frac{4x}{\left(x-1\right)\left(x+1\right)}.\frac{\left(x-1\right)\left(x+1\right)}{1-x^2}=\frac{4x}{1-x^2}\)
b/ Ta có \(3+2\sqrt{2}=\left(\sqrt{2}+1\right)^2\Rightarrow\sqrt{3+\sqrt{8}}=\sqrt{2}+1\)
Suy ra : Nếu x = \(\sqrt{2}+1\) thì \(A=\frac{4\left(\sqrt{2}+1\right)}{1-\left(\sqrt{2}+1\right)^2}=\frac{4\left(\sqrt{2}+1\right)}{-\sqrt{2}.\sqrt{2}\left(\sqrt{2}+1\right)}=-\frac{4}{2}=-2\)
c/ \(A=\sqrt{5}\Rightarrow4x=\sqrt{5}\left(1-x^2\right)\Leftrightarrow\sqrt{5}x^2+4x-\sqrt{5}=0\)
Nhân cả hai vế của pt trên với \(\sqrt{5}\ne0\)
Được \(5x^2+4\sqrt{5}x-5=0\) . Đặt \(t=x\sqrt{5}\) pt trở thành \(t^2+4t-5=0\Leftrightarrow\left(t+5\right)\left(t-1\right)=0\) \(\Leftrightarrow\left[\begin{array}{nghiempt}t=1\\t=-5\end{array}\right.\)
Với t = 1 thì \(x=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}\)
Với t = -5 thì \(x=-\frac{5}{\sqrt{5}}=-\sqrt{5}\)
\(A=\left[\frac{x^2+2x+1-x^2+2x-1}{x^2-1}\right]:\left[\frac{2-x^2-x+x-1}{x^2-1}\right]=\left[\frac{4x}{x^2-1}\right].\left[\frac{x^2-1}{1-x^2}\right]=\frac{4x}{1-x^2}\)
1) Nhìn cái pt hết ham, nhưng bấm nghiệm đẹp v~`~
\(\left(\sqrt{2}+2\right)\left(x\sqrt{2}-1\right)=2x\sqrt{2}-\sqrt{2}\)
\(\Leftrightarrow\left(\sqrt{2}+2\right)\left(x\sqrt{2}-1\right)-2x\sqrt{2}+\sqrt{2}=0\)
\(\Leftrightarrow2x-\sqrt{2}+2x\sqrt{2}-2-2x\sqrt{2}+\sqrt{2}=0\)
\(\Leftrightarrow2x-2=0\Leftrightarrow2x=2\Rightarrow x=1\)
a: \(A=\dfrac{1}{\sqrt{x}+1}:\left(\dfrac{x-9-x+4+\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\right)\)
\(=\dfrac{1}{\sqrt{x}+1}\cdot\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}{\sqrt{x}-3}\)
\(=\dfrac{\sqrt{x}-2}{\sqrt{x}+1}\)
b: Để A<0 thì \(\sqrt{x}-2< 0\)
hay 0<x<4
\(A=\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}+\frac{x^2-4x-1}{x^2-1}\right)\div\frac{x}{x+2019}\)
ĐK : x ≠ ±1 ; x ≠ 0 ; x ≠ -2019
\(=\left(\frac{\left(x+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}-\frac{\left(x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{x^2-4x-1}{\left(x-1\right)\left(x+1\right)}\right)\times\frac{x+2019}{x}\)
\(=\left(\frac{x^2+2x+1}{\left(x-1\right)\left(x+1\right)}-\frac{x^2-2x+1}{\left(x-1\right)\left(x+1\right)}+\frac{x^2-4x-1}{\left(x-1\right)\left(x+1\right)}\right)\times\frac{x+2019}{x}\)
\(=\left(\frac{x^2+2x+1-x^2+2x-1+x^2-4x-1}{\left(x-1\right)\left(x+1\right)}\right)\times\frac{x+2019}{x}\)
\(=\frac{x^2-1}{x^2-1}\times\frac{x+2019}{x}=\frac{x+2019}{x}\)
b. \(A=\frac{x+2019}{x}=1+\frac{2019}{x}\) đạt giá trị lớn nhất
<=> \(\frac{2019}{x}\) đạt giá trị lớn nhất
<=> \(\hept{\begin{cases}x>0\\x\in Z\end{cases}}\) và x đạt giá trị bé nhất
<=> x = 1
Khi đó A = 2020