Question

The minimum value of the expression $y=|x-2|+|x+4|+|x-6|$ is

• A. $6$
• B. $8$
• C. $10$
• D. $14$

Answer 1

The correct option is C $10$

$y=|x-2|+|x+4|+|x-6|$
Here the turning points are $-4,2,6$

$|x-2|+|x+4|+|x-6|=\{\begin{array}{cc}-(x-2)-(x+4)-(x-6),& x<-4\\ -(x-2)+(x+4)-(x-6),& -4\le x<2\\ (x-2)+(x+4)-(x-6),& 2\le x<6\\ (x-2)+(x+4)+(x-6),& x\ge 6\end{array}$

$\Rightarrow |x-2|+|x+4|+|x-6|=\{\begin{array}{cc}4-3x,& x<-4\\ 12-x,& -4\le x<2\\ 8+x,& 2\le x<6\\ 3x-4,& x\ge 6\end{array}$


Hence, the minimum value of the expression is $10$.