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Giaỉ phương trình:
\( \sqrt{x-2009}-1/{x-2009}+ \sqrt{y-2010}-1/y-2010+ \sqrt{z-2011}-1/z-2011 =3/4\)
−1x−2009+y−2010−−−−−−−√−1y−2010+z−2011−−−−−−−√−1z−2011=34
Ta có
x−2009−−−−−−−√−1x−2009+y−2010−−−−−−−√−1y−2010+z−2011−−−−−−−√−1z−2011=34⇔(1x−2009−−−−−−−√−12)2+(1y−2010−−−−−−−√−12)2+(1z−2011−−−−−−−√−12)2=0
⇒x=2013,y=2014,z=2015
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
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
Thưa bn mk đã làm ra nhưng không biết có đúng không. Xem nhá:
Ta có:
\(\frac{\sqrt{x-2009}-1}{x-2009}+\frac{\sqrt{y-2001}-1}{y-2001}+\frac{\sqrt{z-2011}-1}{z-2011}=\frac{3}{4}\Leftrightarrow"\frac{1}{\sqrt{x-2009}}-\frac{1}{2}"^2+\)
\("\frac{1}{\sqrt{y-2010}}-\frac{1}{2}"^2-"\frac{1}{\sqrt{z-2011}}-\frac{1}{2}"^2=0\)
\(\Rightarrow x=2013;y=2014;z=2015\)
P/s: Bn thay Ngoặc Kép thành Ngoặc Đơn nhé
\(\dfrac{\sqrt{x-2009}-1}{x-2009}+\dfrac{\sqrt{y-2010}-1}{y-2010}+\dfrac{\sqrt{z-2011}-1}{z-2011}=\dfrac{3}{4}\)\(\left(\left\{{}\begin{matrix}x>2009\\y>2010\\z>2011\end{matrix}\right.\right)\)
\(\Leftrightarrow\dfrac{1}{4}-\dfrac{\sqrt{x-2009}-1}{x-2009}+\dfrac{1}{4}-\dfrac{\sqrt{y-2010}-1}{y-2010}+\dfrac{1}{4}-\dfrac{\sqrt{z-2011}-1}{z-2011}=0\)
\(\Leftrightarrow\dfrac{x-2009-4\sqrt{x-2009}+4}{x-2009}+\dfrac{y-2010-4\sqrt{y-2010}+4}{y-2010}+\dfrac{z-2011-4\sqrt{z-2011}+4}{z-2011}=0\)
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 \(\left(x;y;z\right)=\left(2013;2014;2015\right)\)
\(\Leftrightarrow\dfrac{4\sqrt{x-2009}-4}{x-2009}-1+\dfrac{4\sqrt{x-2009}-4}{x-2009}-1+\dfrac{4\sqrt{x-2009}-4}{x-2009}-1=0\)\(\Leftrightarrow-\dfrac{\left(\sqrt{x-2009}-2\right)^2}{x-2009}-\dfrac{\left(\sqrt{y-2010}-2\right)^2}{y-2010}-\dfrac{\left(\sqrt{z-2011}-2\right)^2}{z-2011}=0\)
VT <=0 đẳng thức khi và chỉ khi \(\left\{{}\begin{matrix}x-2009=4=>x=2013\\y=2014\\z=2015\end{matrix}\right.\)
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
Lời giải:
Ta có $$\frac{\sqrt{x-2009}-1}{x-2009}+\frac{\sqrt{y-2010}-1}{y-2010}+\frac{\sqrt{z-2011}-1}{z-2011}=\frac{3}{4} \Leftrightarrow \left ( \frac{1}{\sqrt{x-2009}}-\frac{1}{2} \right )^2+\left ( \frac{1}{\sqrt{y-2010}}-\frac{1}{2} \right )^2+\left ( \frac{1}{\sqrt{z-2011}}-\frac{1}{2} \right )^2=0$$
$$\Rightarrow x=2013,y=2014,z=2015$$ 
\(C=\sqrt[3]{2011}-\sqrt[3]{2010}=\frac{2011-2010}{\left(\sqrt[3]{2011^2}+\sqrt[3]{2011}\sqrt[3]{2010}+\sqrt[3]{2010^2}\right)}=\frac{1}{\left(\sqrt[3]{2011^2}+\sqrt[3]{2011}\sqrt[3]{2010}+\sqrt[3]{2010^2}\right)}\)
\(B=\sqrt[3]{2010}-\sqrt[3]{2009}=\frac{2010-2009}{\left(\sqrt[3]{2010^2}+\sqrt[3]{2010}\sqrt[3]{2009}+\sqrt[3]{2009^2}\right)}=\frac{1}{\left(\sqrt[3]{2010^2}+\sqrt[3]{2010}\sqrt[3]{2009}+\sqrt[3]{2009^2}\right)}\)Vì \(\left(\sqrt[3]{2011^2}+\sqrt[3]{2011}\sqrt[3]{2010}+\sqrt[3]{2010^2}\right)>\left(\sqrt[3]{2010^2}+\sqrt[3]{2010}\sqrt[3]{2009}+\sqrt[3]{2009^2}\right)\)
\(B< C\)
lập phương B , C lên