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Giải phương trình chứ chứng minh cái gì
\(\frac{1}{2x-2006}+\frac{1}{3-2007x}+\frac{1}{2006x+2005}=\frac{1}{x+2}\)
\(\Leftrightarrow\left(\frac{1}{2x-2006}-\frac{1}{x+2}\right)+\left(\frac{1}{3-2007x}+\frac{1}{2006x+2005}\right)=0\)
\(\Leftrightarrow\frac{x-2008}{\left(2x-2006\right)\left(x+2\right)}+\frac{x-2008}{\left(3-2007x\right)\left(2006x-2005\right)}=0\)
\(\Leftrightarrow\left(x-2008\right)\left(\frac{1}{\left(2x-2006\right)\left(x+2\right)}+\frac{1}{\left(3-2007x\right)\left(2006x-2005\right)}\right)=0\)
\(\Leftrightarrow\left(x-2008\right)\left(2008x-1\right)\left(2005x+2003\right)=0\)
\(\Leftrightarrow x=2008;x=\frac{1}{2008};x=-\frac{2003}{2005}\)
\(A=\dfrac{x^2-2x+2007}{2007x^2}=\dfrac{2006}{2007^2}+\dfrac{x^2-4014x+2007^2}{2007^2x^2}=\dfrac{2006}{2007^2}+\dfrac{\left(x-2007\right)^2}{2007^2x^2}\ge\dfrac{2006}{2007^2}\)
Vậy GTNN là \(A=\dfrac{2006}{2007^2}\) đạt được khi \(x=2007\)
a) \(x^2+7x+6\)
\(=x^2+x+6x+6\)
\(=x\left(x+1\right)+6\left(x+1\right)\)
\(=\left(x+1\right)\left(x+6\right)\)
b) \(x^4 +2008.x^2+2007.x+2008\)
\(= x^4 +2008x^2+2008x-x+2008\)
\(= x(x^3-1)+2008(x^2+x+1) \)
\(= x(x-1)(x^2+x+1)+2008(x^2+x+1) \)
\(= (x^2+x+1)(x^2-x+2008) \)
\(A=\frac{2007x^2-2x.2007+2007^2}{2007x^2}=\frac{x^2-2x.2007+2007^2}{2007x^2}+\frac{2006x^2}{2007x^2}\)
\(=\frac{\left(x-2007\right)^2}{2007x^2}+\frac{2006}{2007}\ge\frac{2006}{2007}\)
A min =\(\frac{2006}{2007}\)khi \(x-2007=0\)
\(\Leftrightarrow x=2007\)
\(A=\frac{2007x^2-2x.2007+2007^2}{2007x^2}\)
\(A=\frac{x^2-2x.2007-2007^2}{2007x^2}+\frac{2006x^2}{2007x^2}\)
\(A=\frac{\left(x-2007\right)^2}{2007x^2}+\frac{2006}{2007}\ge\frac{2006}{2007}\)
\(\Rightarrow Amin=\frac{2006}{2007}\)khi \(x-2007=0\)
\(\Rightarrow x=2007\)
x=2006
=>x+1=2007
thay x+1=2007 vào A ta được:
A=x6-(x+1)x5+(x+1)x4-(x+1)x3+(x+1)x2-(x+1)x+(x+1)
=x6-x6-x5+x5+x4-x4-x3+x3+x2-x2-x+x+1
=1
Vậy với x=2006 thì A=1
Thay x=2006 vào đa thức A,ta có:
A=20066-2007.20065+2007.20064-2007.20063+2007.20062-2007.2006+2007
=20066-(2006+1).20065+(2006+1).20064-(2006+1).20063+(2006+1).20062-(2006+1).2006+(2006+1)
=20066-20066-20065+20065+20064-20064-20063+20063+20062-20062-2006+2006+1
=(20066-20066)+(-20065+20065)+(20064-20064)+(-20063+20063)+(20062-20062)+(-2006+2006)+1
=1