Question

The minimum value of $f\left(x\right)=|3-x|+|2+x|+|5-x|$ is

• A. $0$
• B. $7$
• C. $8$
• D. $10$

Answer 1

The correct option is B $7$

f can be written as

$f\left(x\right)=\{\begin{array}{c}(3-x)-(2+x)+5-x\text{if}x\le -2\\ (3-x)+(2+x)+5-x\text{if}-2<x<3\\ -(3-x)+(2+x)+5-x\text{if}3\le x<5\\ -(3-x)+(2+x)-(5-x)\text{if}x\ge 5\end{array}$

$f\left(x\right)=\{\begin{array}{c}6-3x\text{if}x\le -2\\ 10-x\text{if}-2<x<3\\ x+4\text{if}3\le x<5\\ 3x-6\text{if}x\ge 5\end{array}$

By drawing the graph of f(x) we see that the minimum value of f is 7.