Question $2(20$ pts $.)$: Three routes connect an origin and destination with link performance functions $t_1=2+0\mathrm{.}5x_1,t_2=1+x_2,$ and $t_3=4+0\mathrm{.}2x_3($with t$\text{'}$s in minutes and x$\text{'}$s in thousands of vehicles per hour$).$

a$.$ Show that when the traffic demand between the origin and destination is $500$ vehicles per hour, no vehicles will take route $1$ and route $3.$

b$.$ What routes will be used if the traffic demand between the origin and destination is $2,000$ vehicles per hour? Justify your answer.

c$.$ What is the minimum origin$-$to$-$destination traffic demand $($in vehicles per hour$)$ that will ensure that all routes will be used under User Equilibrium $($UE$)$ condition?

d$.$ What are the UE flows on the three routes if the traffic demand is the minimum you found in the previous step $($part c$)$ plus $500$ vehicles?