The sum of all local minimum values of the function $f\left(x\right)=\left\{\begin{array}{cc}1-2x,& x<-1\\ \frac{1}{3}(7+2|x\left|\right),& -1\le x\le 2\\ \frac{11}{18}(x-4)(x-5),& x>2\end{array}\right.$

$f\left(x\right)=\left\{\begin{array}{cc}1-2x,& x<-1\\ \frac{1}{3}(7-2x),& -1\le x\le 0\\ \frac{1}{3}(7+2x)& 0\le x<2\\ \frac{11}{18}(x-4)(x-5),& x>2\end{array}\right.$

$\begin{array}{l}\therefore \text{ Local minimum values at A \& B }\\ \frac{7}{3}-\frac{11}{72}\\ \Rightarrow \frac{168-11}{72}\Rightarrow \frac{157}{72}\end{array}$