smc

Note that at $x=2,3,4,$ one of the three absolute values is equal to $0.$ Without using absolute values, we rewrite $f(x)$ as a piecewise function: $$f(x)=\{\begin{array}{ll}(x-2)+(4-x)-(6-2x)& \mathrm{if}2\le x<3\\ (x-2)+(4-x)-(2x-6)& \mathrm{if}3\le x<4\\ (x-2)+(x-4)-(2x-6)& \mathrm{if}4\le x\le 8\end{array},$$ which simplifies to $$f(x)=\{\begin{array}{ll}2x-4& \mathrm{if}2\le x<3\\ -2x+8& \mathrm{if}3\le x<4\\ 0& \mathrm{if}4\le x\le 8\end{array}.$$ The graph of $y=f(x)$ is shown below. The largest value of $f(x)$ is $2,$ and the smallest value of $f(x)$ is $0.$ So, their sum is $\boxed{2}.$