Consider the following mixed$-$integer linear program.

Max $3$x$1+4$x$2$

s$.$t$.$

$4$x$1+9$x$2<=36$

$9$x$1+7$x$2<=63$

x$1,$ x$2>=$ and x$1$ integer

$($a$)$

Graph the constraints for this problem. Indicate on your graph all feasible mixed$-$integer solutions.

On the coordinate plane the horizontal axis is labeled x$1$ and the vertical axis is labeled x$2.$ A region bounded by a series of connected line segments, and several horizontal lines are on the graph.

The series of line segments connect the approximate points $(0,4),(5\mathrm{.}943,1\mathrm{.}358),$ and $(7,0).$

The region is above the horizontal axis, to the right of the vertical axis, and below the line segments.

At each integer value between $0$ and $4$ on the vertical axis, a horizontal line extends out from the vertical axis to the series of connect line segments.

On the coordinate plane the horizontal axis is labeled x$1$ and the vertical axis is labeled x$2.$ A region bounded by a series of connected line segments, and several horizontal lines are on the graph.

The series of line segments connect the approximate points $(0,7),(1\mathrm{.}358,5\mathrm{.}943),$ and $(4,0).$

The region is above the horizontal axis, to the right of the vertical axis, and below the line segments.

At each integer value between $0$ and $7$ on the vertical axis, a horizontal line extends out from the vertical axis to the series of connect line segments.

On the coordinate plane the horizontal axis is labeled x$1$ and the vertical axis is labeled x$2.$ A region bounded by a series of connected line segments, and several vertical lines are on the graph.

The series of line segments connect the approximate points $(0,4),(5\mathrm{.}943,1\mathrm{.}358),$ and $(7,0).$

The region is above the horizontal axis, to the right of the vertical axis, and below the line segments.

At each integer value between $0$ and $7$ on the horizontal axis, a vertical line extends up from the horizontal axis to the series of connect line segments.

On the coordinate plane the horizontal axis is labeled x$1$ and the vertical axis is labeled x$2.$ A region bounded by a series of connected line segments, and several vertical lines are on the graph.

The series of line segments connect the approximate points $(0,7),(1\mathrm{.}358,5\mathrm{.}943),$ and $(4,0).$

The region is above the horizontal axis, to the right of the vertical axis, and below the line segments.

At each integer value between $0$ and $4$ on the horizontal axis, a vertical line extends up from the horizontal axis to the series of connect line segments.

$($b$)$

Find the optimal solution to the LP Relaxation. Round the value of

x$1$

down to find a feasible mixed$-$integer solution. $($Round the values of

x$2$

and the optimal solution to three decimal places, if necessary.$)$

at

$($x$1,$ x$2)=$

Is this solution optimal?

Yes

No

$($c$)$

Find the optimal solution for the mixed$-$integer linear program. $($Round your answers to three decimal places, when necessary.$)$

at

$($x$1,$ x$2)=$