$9-5$ Consider the following graph.

$($a$)$ Find $($by inspection$)$ shortest paths from all nodes to all other nodes.

$($b$)$ Verify that every subpath of the optimal $1$ to $4$ path in part $($a$)$ is itself optimal.

$($c$)$ Detail your optimal solutions of part $($a$)$ in functional notation

$u[k][l]$ and $x_{i,j}[k][l].$

$($d$)$ Write functional equations for the shortest path problems of part $($a$).$

$($e$)$ Verify that the

$u[k][l]$ of part $($c$)$ satisfy functional equations of part $($d$).$