Question

[ha nguyen thi]

tìm GtNN : A = I x - 1 I + I x -2 I + Ix - 3 I + Ix - 4 I + 15

Answer 1

[Trang Nguyễn]

Đặt `B = |x - 1| + |x - 2| + |x - 3| + |x - 4|`

`= (|x - 1| + |x - 4|) + (|x - 2| + |x - 3|)`

`= (|x - 1| + |4 - x|) + (|x - 2| + |3 - x|)`

\(\Rightarrow B\ge\left|x-1+4-x\right|+\left|x-2+3-x\right|\)

\(B\ge\left|3\right|+\left|1\right|=4\)

\(\Rightarrow A\ge4+15=19\)

hay MinA = 19

Dấu "=" xảy ra khi: \(\left\{{}\begin{matrix}\left(x-1\right)\left(4-x\right)\ge0\\\left(x-2\right)\left(3-x\right)\ge0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\left(x-1\right)\left(x-4\right)\le0\\\left(x-2\right)\left(x-3\right)\le0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}1\le x\le4\\2\le x\le3\end{matrix}\right.\Rightarrow2\le x\le3\)

Vậy MinA = 19 tại \(2\le x\le3\).