Let's walk through the questions presented in the image.

3.1 (9 pts) Completing the Inequalities

We are asked to complete the inequalities for three lines (A, B, and C) based on the graph and the feasible region.

• Line A: The equation is $y = -2$, which is a horizontal line. The feasible region is above this line, so the inequality should be: $y \geq -2$

• Line B: The equation of the line is $y = x + 3$. The feasible region lies below the line, so the inequality should be: $y \leq x + 3$

• Line C: The equation is $y = -2x + 6$. The feasible region lies below this line, so the inequality should be: $y \leq -2x + 6$

3.2 (8 pts) Maximum of the Objective Function

The objective function is $F = -3x + 2y$. To check if there's a maximum, we need to evaluate $F$ at the vertices of the feasible region (since linear programming problems have their extrema at the vertices of the feasible region).

The vertices of the feasible region are:

• $(-1, 2)$
• $(4, 2)$
• $(4, -2)$

Now, we'll compute $F$ at each vertex:

• At $(-1, 2)$: $F = -3(-1) + 2(2) = 3 + 4 = 7$

• At $(4, 2)$: $F = -3(4) + 2(2) = -12 + 4 = -8$

• At $(4, -2)$: $F = -3(4) + 2(-2) = -12 - 4 = -16$

The maximum value of $F$ occurs at $(-1, 2)$ with $F = 7$.

3.3 (8 pts) Minimum of the Objective Function

We already calculated $F$ at the vertices of the feasible region. The minimum value is $-16$, which occurs at $(4, -2)$.

3.4 (8 pts) Objection Function for Minimum at (1, 4)

We need to find an objective function for which the vertex $(1, 4)$ is the minimum. One approach is to choose a linear function that decreases in the direction of this point relative to the other vertices of the feasible region.

An example of such a function could be: $F = 2x + y$

To confirm, we calculate $F$ at the relevant vertices (including $(1, 4)$) and show that $(1, 4)$ gives the smallest value:

• At $(1, 4)$: $F = 2(1) + 4 = 2 + 4 = 6$

• At another vertex (to compare), for instance $(0, 2)$: $F = 2(0) + 2 = 0 + 2 = 2$

This approach shows how you can create an objective function to minimize at $(1, 4)$.

Do you want more details or have any questions on these problems? Here are 5 related questions to further explore:

1. How do you graph linear inequalities and identify feasible regions?
2. What methods are used to find maximum and minimum values in linear programming?
3. Can you explain why linear programming solutions always occur at the vertices of the feasible region?
4. How would the problem change if the objective function was different?
5. How can we test if a solution is optimal in linear programming?

Tip: Always evaluate the objective function at all vertices of the feasible region to find maximum or minimum values in linear programming problems.