$12\mathrm{.}8-2.*$ The Dorwyn Company has two new products that will compete with the two new products for the Wyndor Glass Co$.($described in Sec. $3\mathrm{.}1).$ Using units of hundreds of dollars for the objective function, the linear programming model shown below has been formulated to determine the most profitable product mix.

Maximize $,Z=4x_1+6x_2$

subject to

$x_1+3x_{2}8$

$5x_1+2x_{2}14$

and

$x_{1}0,x_{2}0$

However, because of the strong competition from Wyndor, Dorwyn management now realizes that the company will need to make a strong marketing effort to generate substantial sales of these products. In particular, it is estimated that achieving a production and sales rate of $x_1$ units of Product $1$ per week will require weekly marketing costs of $x_1^3$ hundred dollars. The corresponding marketing costs for Product $2$ are estimated to be $2x_2^2$ hundred dollars. Thus, the objective function in the model should be $Z=4x_1+6x_2-x_1^3-2x_2^2.$

Dorwyn management now would like to use the revised model to determine the most profitable product mix.

$($a$)$ Verify that $(x_1,x_2)=(\frac{2}{\sqrt[\phantom{2}]{3}},\frac{3}{2})$ is an optimal solution by applying the KKT conditions.

$($b$)$ Construct tables to show the profit data for each product when the production rate is $0,1,2,3.$

$($c$)$ Draw a figure like Fig. $13\mathrm{.}15b$ that plots the weekly profit points for each product when the production rate is $0,1,2,3.$ Connect the pairs of consecutive points with $($dashed$)$ line segments.

$($d$)$ Use separable programming based on this figure to formulate an approximate linear programming model for this problem.