Let x$3$ and x$4$ denote the slack variables for the respective functional constraints. When c$1$

$=3,$ c$2=-2,$ b$1=30,$ and b$2=10,$ the simplex method yields the following final simplex

tableau.

$($a$)$ Use graphical analysis to determine the allowable range for c$1$ and c$2.$

$($b$)$ Use algebraic analysis to derive and verify your answers in part $($a$).$

$($c$)$ Use graphical analysis to determine the allowable range for b$1$ and b$2.$

$($d$)$ Use algebraic analysis to derive and verify your answers in part $($c$)$