Question

The maximum value of $y=|x-4|-|x-7|$ is

Answer 1

$y=|x-4|-|x-7|$
Here the turning points are $4,7$
$y=\{\begin{array}{cc}-(x-4)+(x-7)& x<4\\ (x-4)+(x-7)& 4\le x<7\\ (x-4)-(x-7)& x\ge 7\end{array}\Rightarrow y=\{\begin{array}{cc}-3& x<4\\ 2x-11& 4\le x<7\\ 3& x\ge 7\end{array}$


Hence, the maximum value is $3$.