Question

Cho \(x\in R\). Tìm GTNN của hàm số:

\(f\left(x\right)=\left|x-1\right|+2\left|x-2\right|+3\left|x-3\right|+4\left|x-4\right|\)

Answer 1

\(f\left(x\right)=\left|x-1\right|+\left|4-x\right|+2\left(\left|x-2\right|+\left|4-x\right|\right)+\left|x-3\right|+\left|4-x\right|+2\left|x-3\right|\)

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

\(f\left(x\right)\ge3+4+1+2\left|x-3\right|=8+2\left|x-3\right|\ge8\)

\(\Rightarrow f\left(x\right)_{min}=8\) khi \(x=3\)