The problem asks to find the minimum and maximum values of the objective function, given certain constraints. Here is a breakdown of the problem:

Objective Function:

$f(x, y) = 3x + 14y$

Constraints:

1. $x \geq 0$
2. $y \geq 0$
3. $13x + 5y \leq 130$
4. $13x + 5y \leq 260$

This is a linear programming problem, and the solution involves:

1. Identifying the feasible region determined by the constraints.
2. Finding the values of $f(x, y)$ at the corner points (vertices) of the feasible region.
3. Choosing the minimum and maximum values from the evaluated points.

Step 1: Set up the inequalities and solve for the corner points.

1. For $13x + 5y = 130$:

   • When $x = 0$, solve for $y$: $5y = 130 \implies y = 26$
   • When $y = 0$, solve for $x$: $13x = 130 \implies x = 10$ So, the line passes through the points $(0, 26)$ and $(10, 0)$.

2. For $13x + 5y = 260$:

   • When $x = 0$, solve for $y$: $5y = 260 \implies y = 52$
   • When $y = 0$, solve for $x$: $13x = 260 \implies x = 20$ So, the line passes through the points $(0, 52)$ and $(20, 0)$.

Step 2: Identify the corner points of the feasible region.

Since both inequalities must hold, the corner points can be found by intersecting the lines and considering the constraints $x \geq 0$ and $y \geq 0$. The corner points are:

• $(0, 26)$
• $(10, 0)$
• $(0, 52)$
• $(20, 0)$

Step 3: Evaluate the objective function at the corner points.

For each corner point $(x, y)$, calculate $f(x, y)$:

1. At $(0, 26)$: $f(0, 26) = 3(0) + 14(26) = 0 + 364 = 364$
2. At $(10, 0)$: $f(10, 0) = 3(10) + 14(0) = 30 + 0 = 30$
3. At $(0, 52)$: $f(0, 52) = 3(0) + 14(52) = 0 + 728 = 728$
4. At $(20, 0)$: $f(20, 0) = 3(20) + 14(0) = 60 + 0 = 60$

Step 4: Identify the minimum and maximum values.

• Minimum value of $f(x, y) = 30$ at $(10, 0)$.
• Maximum value of $f(x, y) = 728$ at $(0, 52)$.

Final Answer:

• Minimum value: $30$ at $(10, 0)$
• Maximum value: $728$ at $(0, 52)$

Would you like to explore any of the steps further or have questions on how the solution was derived?

Follow-up Questions:

1. What happens if one of the constraints is changed?
2. Can you use graphical methods to visualize this solution?
3. How do changes in the objective function coefficients affect the optimal points?
4. What are the conditions for the problem to have an unbounded solution?
5. Can this method be extended to problems with more than two variables?

Tip:

In linear programming, always check the feasibility of corner points, as optimal solutions typically occur there when dealing with two-variable problems.