Đáp án:
`M = |x + 8| + |x + 4| + |x + 2018|`
`⇔ M = |x + 8| + |x + 2018| + |x + 4|`
`⇔ M = |x + 8| + |-x - 2018| + |x + 4|`
Áp dụng BĐT `|a| + |b| ≥ |a + b|` có :
`|x + 8| + |-x - 2018| + |x+ 4| ≥ |x + 8 - x - 2018| + |x + 4| = 2010 + |x + 4|`
`-> M ≥ 2010`
`-> M_{min} = 2010`
Dấu "`=`" xảy ra khi :
\(\left\{ \begin{array}{l}(x + 8) (-x - 2018)≥0\\x + 4= 0 \end{array} \right.\) `⇔` \(\left\{ \begin{array}{l}(x + 8) (-x - 2018)≥0\\x = -4\end{array} \right.\)
Với `(x + 8) (-x - 2018) ≥ 0`
`⇔` \(\left\{ \begin{array}{l} x + 8≥0\\-x-2018≤0\end{array} \right.\) `⇔` \(\left\{ \begin{array}{l}x≥-8\\x≤-2018\end{array} \right.\) `⇔ -8≤x≤-2018` (Vô lí)
`⇔` \(\left\{ \begin{array}{l}x+8≤0\\-x-2018 ≥0\end{array} \right.\) `⇔` \(\left\{ \begin{array}{l}x≤-8\\x≥-2018\end{array} \right.\) `⇔ -2018 ≤ x ≤ -8` (Thỏa mãn) `⇔ x = -4`
Vậy `M_{min} = 2010 ⇔ x = -4`
