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Thay \(x=2003\) vào A ta có:\(A=2003^{17}-2004.2003^{16}+2004.2003^{15}-2004.2003^{14}+...+2004.\left(2003-1\right)\)
\(=2003^{17}-\left(2003+1\right).2003^{16}+\left(2003+1\right).2003^{15}-\left(2003+1\right).2003^{14}+...+\left(2003+1\right).\left(2003-1\right)\)
\(=2003^{17}-2003^{17}+2003^{16}-2003^{16}+2003^{15}-2003^{15}+2003^{14}-2003^{14}+...+\left(2003+1\right).\left(2003-1\right)\)
\(=2004.2002=4012008\)
Với x = 2005 ta có
\(x^{2005}-2006x^{2004}+2006x^{2003}-2006x^{2002}+...-2006x^2+2006x-1\)
\(=\left(x^{2005}-2005x^{2004}\right)-\left(x^{2004}-2005^{2003}\right)+\left(x^{2003}-2005x^{2002}\right)-...-\left(x^2-2005x\right)+\left(x-2005\right)+2006\)
\(=\left(x-2005\right)\left(x^{2004}-x^{2003}+x^{2002}-...-x+1\right)+2006=2006\).
Ta có : \(\left|2004-x\right|+\left|2003-x\right|\)
\(\Rightarrow\left|2004-x\right|+\left|2003-x\right|\ge\left|2004-x+x-2003\right|=1\)
Dấu \("="\) xảy ra \(\Leftrightarrow\left(2004-x\right).\left(x-2003\right)\ge0\)
\(\Rightarrow\left[\begin{array}{nghiempt}\hept{\begin{cases}2004-x\ge0\\x-2003\ge0\end{array}\right.\\\hept{\begin{cases}2004-x\le0\\x-2003\le0\end{array}\right.\end{array}\right.\)
\(\Rightarrow\left[\begin{array}{nghiempt}\hept{\begin{cases}x\le2004\\x\ge2003\end{array}\right.\\\hept{\begin{cases}x\ge2004\\x\le2003\end{array}\right.\end{array}\right.\)
\(\Rightarrow2003\le x\le2004\)
Vậy : Giá trị nhỏ nhất của \(D=1\Leftrightarrow2003\le x\le2004\)
Ta có :
\(x=2005\Rightarrow x+1=2006\)
Thay \(2006=x+1\) vào biểu thức trên ta được :
\(x^{2005}-\left(x+1\right)x^{2004}+\left(x+1\right)x^{2003}-\left(x+1\right)x^{2002}+...-\left(x+1\right)x^2+\left(x+1\right)x-1\)
\(=x^{2005}-x^{2005}+x^{2004}-x^{2004}+x^{2003}-...-x^3+x^2-x^2+x-1\)
\(=x-1\) mà \(x=2005\)
\(\Rightarrow x^{2005}-2006.x^{2004}+2006.x^{2003}-2006.x^{2002}+...-2006.x^2+2006x-1=2005-1=2004\)
f(x)=\(x^{17}-2004.x^{16}+2004.x^{15}-2004.x^{2014}+...+2004.x-1\)
= \(x^{17}-\left(2003+1\right)x^{16}+\left(2003+1\right)x^{15}-\left(2003+1\right)^{14}+...+\left(2003+1\right)-1\)
Thay x = 2003
=> f(x)= \(x^{17}-\left(x+1\right)x^{16}+\left(x+1\right)x^{15}-\left(x+1\right)x^{14}+...+\left(x+1\right)x-1\)
=\(x^{17}-x^{17}-x^{16}+x^{16}+x^{15}-x^{15}-x^{14}+...+x^2+x-1\)
= \(x-1\)
= 2003 -1
=2002