LP Maximization Sensitivity $(15$ points$).$ Recall the following linear program from

Question $1$ on Homework #$3$:

max $3$A $+4$B

s$.$t$.$

$$A $+2$B $8$ Constraint $1$

A $+2$B $12$ Constraint $2$

$2$A $+$ B $16$ Constraint $3$

A$,$ B $^30$ Nonnegativity

The optimal solution to this problem is $($A$*,$ B$*)=(6\mathrm{.}6667,2\mathrm{.}6667),$ which is the extreme

point that is determined by the intersection of Constraint $2$ and Constraint $3.$ Answer the

following questions manually and show your work:

a$)$ Holding the coefficient of $$B$$ in the objective function fixed, how much can the

coefficient of $$A$$ decrease or increase so that the optimal solution does not change?

b$)$ Holding the coefficient of $$A$$ in the objective function fixed, how much can the

coefficient of $$B$$ decrease or increase so that the optimal solution does not change?

c$)$ Suppose the coefficient of $$A$$ changes from $3$ to $5$ and the coefficient of $$B$$

changes from $4$ to $2$ in the objective function. Will the optimal solution change? If so$,$

what is the new optimal solution?