\(\dfrac{\sqrt{x-2009}-1}{x-2009}+\dfrac{\sqrt{y-2010}-1}{y-2010}+\dfrac{\sqrt{z-2011}-1}{z-2011}=\dfrac{3}{4}\)
giải pt
K
Khách
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1 tháng 3 2018
\(\Leftrightarrow\dfrac{4\sqrt{x-2009}-4}{x-2009}-1+\dfrac{4\sqrt{x-2009}-4}{x-2009}-1+\dfrac{4\sqrt{x-2009}-4}{x-2009}-1=0\)\(\Leftrightarrow-\dfrac{\left(\sqrt{x-2009}-2\right)^2}{x-2009}-\dfrac{\left(\sqrt{y-2010}-2\right)^2}{y-2010}-\dfrac{\left(\sqrt{z-2011}-2\right)^2}{z-2011}=0\)
VT <=0 đẳng thức khi và chỉ khi \(\left\{{}\begin{matrix}x-2009=4=>x=2013\\y=2014\\z=2015\end{matrix}\right.\)
1 tháng 5 2018
Đặt a = \(\sqrt{x-2009}\)
b = \(\sqrt{y-2010}\)
c = \(\sqrt{z-2011}\)
\(\Leftrightarrow\dfrac{a-1}{a^2}+\dfrac{b-1}{b^2}+\dfrac{c-1}{c^2}=\dfrac{3}{4}\)
\(\Leftrightarrow\dfrac{1}{a}-\dfrac{1}{a^2}+\dfrac{1}{b}-\dfrac{1}{b^2}+\dfrac{1}{c}-\dfrac{1}{c^2}=\dfrac{3}{4}\)
\(\Leftrightarrow\dfrac{1}{a}-\dfrac{1}{a^2}-\dfrac{1}{4}+\dfrac{1}{b}-\dfrac{1}{b^2}-\dfrac{1}{4}+\dfrac{1}{c}-\dfrac{1}{c^2}-\dfrac{1}{4}=0\)
\(\Leftrightarrow-(\dfrac{1}{a}-\dfrac{1}{2})^2-\left(\dfrac{1}{b}-\dfrac{1}{2}\right)^2-\left(\dfrac{1}{c}-\dfrac{1}{2}\right)^2=0\)
Dấu = xảy ra khi
a = 2
b = 2
c = 2
\(\Leftrightarrow\sqrt{x-2009}=2\)
\(\sqrt{y-2010}=2\)
\(\sqrt{z-2011}=2\)
\(\Leftrightarrow x-2009=4\)
\(y-2010=4\)
\(z-2011=4\)
=> x = 2013
y = 2014
z = 2015
31 tháng 1 2019
Lời giải:
Ta có $$\frac{\sqrt{x-2009}-1}{x-2009}+\frac{\sqrt{y-2010}-1}{y-2010}+\frac{\sqrt{z-2011}-1}{z-2011}=\frac{3}{4} \Leftrightarrow \left ( \frac{1}{\sqrt{x-2009}}-\frac{1}{2} \right )^2+\left ( \frac{1}{\sqrt{y-2010}}-\frac{1}{2} \right )^2+\left ( \frac{1}{\sqrt{z-2011}}-\frac{1}{2} \right )^2=0$$
$$\Rightarrow x=2013,y=2014,z=2015$$ 
G
14 tháng 5 2019
Đặt: \(\hept{\begin{cases}\sqrt{x-2009}=a\\\sqrt{y-2010}=b\\\sqrt{z-2011}=c\end{cases}}\)
Ta có: \(\frac{1}{a}-\frac{1}{a^2}+\frac{1}{b}-\frac{1}{b^2}+\frac{1}{c}-\frac{1}{c^2}-\frac{3}{4}=0\)
\(\Leftrightarrow\frac{1}{a^2}-\frac{1}{a}+\frac{1}{b^2}-\frac{1}{b}+\frac{1}{c^2}-\frac{1}{c}+\frac{3}{4}=0\)
\(\Leftrightarrow\left(\frac{1}{a^2}-\frac{1}{a}+\frac{1}{4}\right)+\left(\frac{1}{b^2}-\frac{1}{b}+\frac{1}{4}\right)+\left(\frac{1}{c^2}-\frac{1}{c}+\frac{1}{4}\right)=0\)
\(\Leftrightarrow\left(\frac{1}{a}-\frac{1}{2}\right)^2+\left(\frac{1}{b}-\frac{1}{2}\right)^2+\left(\frac{1}{c}-\frac{1}{2}\right)^2=0\)
\(\Leftrightarrow a=b=c=\frac{1}{2}\)
Thay vào tìm x;y;z
24 tháng 9 2019
Đặt: \(\hept{\begin{cases}\sqrt{x-2009}=a\\\sqrt{y-2010}=b\\\sqrt{z-2011}=c\end{cases}}\)
Ta có: \frac{1}{a}-\frac{1}{a^2}+\frac{1}{b}-\frac{1}{b^2}+\frac{1}{c}-\frac{1}{c^2}-\frac{3}{4}=0a1−a21+b1−b21+c1−c21−43=0
\Leftrightarrow\frac{1}{a^2}-\frac{1}{a}+\frac{1}{b^2}-\frac{1}{b}+\frac{1}{c^2}-\frac{1}{c}+\frac{3}{4}=0⇔a21−a1+b21−b1+c21−c1+43=0
\Leftrightarrow\left(\frac{1}{a^2}-\frac{1}{a}+\frac{1}{4}\right)+\left(\frac{1}{b^2}-\frac{1}{b}+\frac{1}{4}\right)+\left(\frac{1}{c^2}-\frac{1}{c}+\frac{1}{4}\right)=0⇔(a21−a1+41)+(b21−b1+41)+(c21−c1+41)=0
\Leftrightarrow\left(\frac{1}{a}-\frac{1}{2}\right)^2+\left(\frac{1}{b}-\frac{1}{2}\right)^2+\left(\frac{1}{c}-\frac{1}{2}\right)^2=0⇔(a1−21)2+(b1−21)2+(c1−21)2=0
\Leftrightarrow a=b=c=\frac{1}{2}⇔a=b=c=21
Thay vào tìm x;y;z
AH
Akai Haruma
Giáo viên
11 tháng 12 2018
Lời giải:
Áp dụng BĐT Cô-si ngược dấu:
\(\sqrt{x-2010}=\frac{1}{2}\sqrt{4(x-2010)}\leq \frac{4+(x-2010)}{4}\)
\(\Rightarrow \sqrt{x-2010}-1\leq \frac{4+(x-2010)}{4}-1=\frac{x-2010}{4}\)
\(\Rightarrow \frac{\sqrt{x-2010}-1}{x-2010}\leq \frac{1}{4}\)
Hoàn toàn tương tự với những phân thức còn lại:
\(\Rightarrow \frac{\sqrt{x-2010}-1}{x-2010}+\frac{\sqrt{y-2011}-1}{y-2011}+\frac{\sqrt{z-2012}-1}{z-2012}\leq \frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\)
Dấu "=" xảy ra khi \(\left\{\begin{matrix} x-2010=4\\ y-2011=4\\ z-2012=4\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x=2014\\ y=2015\\ z=2016\end{matrix}\right.\)
7 tháng 11 2015
Đặt \(a=\sqrt{x-2009};b=\sqrt{y-2010};c=\sqrt{z-2011};a>0;b>0;c>0\)
\(Pt\Leftrightarrow\frac{a-1}{a^2}-\frac{1}{4}+\frac{b-1}{b^2}-\frac{1}{4}+\frac{c-1}{c^2}-\frac{1}{4}=0\)
\(\Leftrightarrow\frac{\left(4a^2-a+1\right)}{a^2}+\frac{\left(4b^2-b+1\right)}{b^2}+\frac{\left(4c^2-c+1\right)}{c^2}=0\)
\(\Leftrightarrow\left(\frac{2a-1}{a}\right)^2+\left(\frac{2b-1}{b}\right)^2+\left(\frac{2c-1}{c}\right)^2=0\)
\(\Rightarrow a=b=c=\frac{1}{2}\Rightarrow\sqrt{x-2009}=\frac{1}{2}\Rightarrow x=2009\frac{1}{4}\)
\(\Rightarrow b=\frac{1}{2}\Rightarrow\sqrt{y-2010}=\frac{1}{2}\Rightarrow y=2010\frac{1}{4}\)
\(\Rightarrow c=\frac{1}{2}\Rightarrow\sqrt{z-2011}=\frac{1}{2}\Rightarrow z=2011\frac{1}{4}\)
\(\dfrac{\sqrt{x-2009}-1}{x-2009}+\dfrac{\sqrt{y-2010}-1}{y-2010}+\dfrac{\sqrt{z-2011}-1}{z-2011}=\dfrac{3}{4}\)\(\left(\left\{{}\begin{matrix}x>2009\\y>2010\\z>2011\end{matrix}\right.\right)\)
\(\Leftrightarrow\dfrac{1}{4}-\dfrac{\sqrt{x-2009}-1}{x-2009}+\dfrac{1}{4}-\dfrac{\sqrt{y-2010}-1}{y-2010}+\dfrac{1}{4}-\dfrac{\sqrt{z-2011}-1}{z-2011}=0\)
\(\Leftrightarrow\dfrac{x-2009-4\sqrt{x-2009}+4}{x-2009}+\dfrac{y-2010-4\sqrt{y-2010}+4}{y-2010}+\dfrac{z-2011-4\sqrt{z-2011}+4}{z-2011}=0\)
Nhận xét: \(\left\{{}\begin{matrix}\dfrac{\left(\sqrt{x-2009}-2\right)^2}{x-2009}\ge0\\\dfrac{\left(\sqrt{y-2010}-2\right)^2}{y-2010}\ge0\\\dfrac{\left(\sqrt{z-2011}-2\right)^2}{z-2011}\ge0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\sqrt{x-2009}-2=0\\\sqrt{y-2010}-2=0\\\sqrt{z-2011}-2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=2013\\y=2014\\z=2015\end{matrix}\right.\)
Vậy \(\left(x;y;z\right)=\left(2013;2014;2015\right)\)