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

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

The function f is decreasing for $x\in \left(-\infty ,3\right)$ and increases on $\left(3,\infty \right)$ . Hence, $x=3$ is a point of minimum and min $f\left(x\right)=7$