Find the maximum and minimum values of `f: 2x + y` subject to the constraints :
`x + 3y ge 6, x - 3y le3, 3x + 4y le 24`,
`-3x + 2y le 6, 5x + y ge 5, x, y ge 0`.

Answer

To find the maximum and minimum values of the function \( f = 2x + y \) subject to the given constraints, we will follow these steps:

### Step 1: Identify the Constraints
The constraints given are:
1. \( x + 3y \geq 6 \)
2. \( x - 3y \leq 3 \)
3. \( 3x + 4y \leq 24 \)
4. \( -3x + 2y \leq 6 \)
5. \( 5x + y \geq 5 \)
6. \( x \geq 0 \)
7. \( y \geq 0 \)

### Step 2: Convert Inequalities to Equations
To graph the constraints, we convert each inequality into an equation:
1. \( x + 3y = 6 \)
2. \( x - 3y = 3 \)
3. \( 3x + 4y = 24 \)
4. \( -3x + 2y = 6 \)
5. \( 5x + y = 5 \)

### Step 3: Find Intercepts
For each equation, we will find the x-intercept and y-intercept:
1. For \( x + 3y = 6 \):
- x-intercept: \( (6, 0) \)
- y-intercept: \( (0, 2) \)

2. For \( x - 3y = 3 \):
- x-intercept: \( (3, 0) \)
- y-intercept: \( (0, -1) \) (not relevant since \( y \geq 0 \))

3. For \( 3x + 4y = 24 \):
- x-intercept: \( (8, 0) \)
- y-intercept: \( (0, 6) \)

4. For \( -3x + 2y = 6 \):
- x-intercept: \( (-2, 0) \) (not relevant since \( x \geq 0 \))
- y-intercept: \( (0, 3) \)

5. For \( 5x + y = 5 \):
- x-intercept: \( (1, 0) \)
- y-intercept: \( (0, 5) \)

### Step 4: Graph the Constraints
Plot the lines on a graph and shade the feasible region based on the inequalities:
- For \( x + 3y \geq 6 \), shade above the line.
- For \( x - 3y \leq 3 \), shade below the line.
- For \( 3x + 4y \leq 24 \), shade below the line.
- For \( -3x + 2y \leq 6 \), shade below the line.
- For \( 5x + y \geq 5 \), shade above the line.

### Step 5: Identify the Feasible Region
The feasible region is where all shaded areas overlap, and it must be bounded by the axes \( x \geq 0 \) and \( y \geq 0 \).

### Step 6: Find Corner Points
Determine the corner points of the feasible region by solving the equations of the lines where they intersect:
1. Intersection of \( x + 3y = 6 \) and \( x - 3y = 3 \):
- Solve the system:
- \( x + 3y = 6 \)
- \( x - 3y = 3 \)
- Adding both equations gives \( 2x = 9 \) → \( x = 4.5 \)
- Substitute \( x \) back to find \( y \): \( 4.5 + 3y = 6 \) → \( 3y = 1.5 \) → \( y = 0.5 \)
- Point: \( (4.5, 0.5) \)

2. Intersection of \( 3x + 4y = 24 \) and \( x - 3y = 3 \):
- Solve the system:
- \( 3x + 4y = 24 \)
- \( x - 3y = 3 \)
- Substitute \( x = 3 + 3y \) into the first equation:
- \( 3(3 + 3y) + 4y = 24 \)
- \( 9 + 9y + 4y = 24 \) → \( 13y = 15 \) → \( y = \frac{15}{13} \)
- Substitute back to find \( x \): \( x = 3 + 3(\frac{15}{13}) = \frac{84}{13} \)
- Point: \( \left(\frac{84}{13}, \frac{15}{13}\right) \)

3. Other intersections can be calculated similarly.

### Step 7: Evaluate the Objective Function
Evaluate \( f = 2x + y \) at each corner point:
1. At \( (6, 0) \): \( f = 2(6) + 0 = 12 \)
2. At \( (8, 0) \): \( f = 2(8) + 0 = 16 \)
3. At \( (4.5, 0.5) \): \( f = 2(4.5) + 0.5 = 9.5 \)
4. At \( \left(\frac{84}{13}, \frac{15}{13}\right) \):
- \( f = 2\left(\frac{84}{13}\right) + \frac{15}{13} = \frac{168 + 15}{13} = \frac{183}{13} \approx 14.08 \)

### Step 8: Determine Maximum and Minimum
- Maximum value occurs at \( (8, 0) \) with \( f = 16 \).
- Minimum value occurs at \( (4.5, 0.5) \) with \( f = 9.5 \).

### Final Result
- Maximum value of \( f \) is \( 16 \) at \( (8, 0) \).
- Minimum value of \( f \) is \( 9.5 \) at \( (4.5, 0.5) \).