\(A=\left|x-2014\right|+\left|x-2015\right|+\left|x-2016\right|\)
\(A=\left|x-2015\right|+\left|x-2014\right|+\left|x-2016\right|\)
\(A=\left|x-2015\right|+(\left|x-2014\right|+\left|x-2016\right|)\)
Áp dụng BĐT \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) ta có:
\(\left|x-2014\right|+\left|x-2016\right|\)
\(=\left|x-2014\right|+\left|2016-x\right|\ge\left|x-2014+2016-x\right|\)
\(=\left|2\right|=2\)
\(\Leftrightarrow\left|x-2014\right|+\left|x-2015\right|+\left|x-2016\right|\ge2\)
Đẳng thức xảy ra \(\Leftrightarrow\left[{}\begin{matrix}x-2014\ge0\\x-2015=0\\x-2016\le0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\ge2014\\x=2015\\x\le2016\end{matrix}\right.\)
\(\Rightarrow Min_A=2\Leftrightarrow x=2015.\)
A = x - 2014 + x - 2015 + x - 2016
A = ( x + x + x ) - ( 2014 + 2015 + 2016 )
A = \(x^3\)- 6045
Vì \(x^3\) có số mũ là số lẻ
\(\rightarrow x^3\)- 6045 = ( - 999.9)-6045
\(\Rightarrow\) A = -100...0006044
A=|x−2014|+|x−2015|+|x−2016|A=|x−2014|+|x−2015|+|x−2016|
A=|x−2015|+|x−2014|+|x−2016|A=|x−2015|+|x−2014|+|x−2016|
A=|x−2015|+(|x−2014|+|x−2016|)A=|x−2015|+(|x−2014|+|x−2016|)
Áp dụng BĐT |a|+|b|≥|a+b||a|+|b|≥|a+b| ta có:
|x−2014|+|x−2016||x−2014|+|x−2016|
=|x−2014|+|2016−x|≥|x−2014+2016−x|=|x−2014|+|2016−x|≥|x−2014+2016−x|
=|2|=2=|2|=2
⇔|x−2014|+|x−2015|+|x−2016|≥2⇔|x−2014|+|x−2015|+|x−2016|≥2
Đẳng thức xảy ra ⇔⎡⎢⎣x−2014≥0x−2015=0x−2016≤0⇔[x−2014≥0x−2015=0x−2016≤0
⇔⎡⎢⎣x≥2014x=2015x≤2016
