Tìm min; max

To solve the optimization problems you've provided, we can break them down step by step:

### 1. Problem a)
Minimize \( F(x, y) = 2x - 3y \) under the constraints:
- \( x \geq 0 \)
- \( y \geq 0 \)
- \( 4x + 2y \leq 8 \)

**Constraints Interpretation:**
1. \( x \geq 0 \)
2. \( y \geq 0 \)
3. From \( 4x + 2y \leq 8 \), we can simplify it to \( 2x + y \leq 4 \).

**Vertices of the feasible region:**
- Intercepts of the line \( 2x + y = 4 \):
- \( x \)-intercept: \( (2, 0) \)
- \( y \)-intercept: \( (0, 4) \)

The vertices of the feasible region are:
- \( (0, 0) \)
- \( (2, 0) \)
- \( (0, 4) \)

**Evaluating the objective function at the vertices:**
- \( F(0, 0) = 2(0) - 3(0) = 0 \)
- \( F(2, 0) = 2(2) - 3(0) = 4 \)
- \( F(0, 4) = 2(0) - 3(4) = -12 \)

**Minimum value:** \( F(0, 4) = -12 \)

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### 2. Problem b)
Minimize \( F(x, y) = x + 5y \) under the constraints:
- \( x \geq 0 \)
- \( y \geq 0 \)
- \( 3x + y \leq 12 \)

**Constraints Interpretation:**
1. \( x \geq 0 \)
2. \( y \geq 0 \)
3. From \( 3x + y \leq 12 \), we plot the line.

**Vertices of the feasible region:**
- \( (0, 0) \)
- \( (0, 12) \) — \( y \)-intercept
- \( (4, 0) \) — \( x \)-intercept

**Evaluating the objective function at the vertices:**
- \( F(0, 0) = 0 + 5(0) = 0 \)
- \( F(0, 12) = 0 + 5(12) = 60 \)
- \( F(4, 0) = 4 + 5(0) = 4 \)

**Minimum value:** \( F(4, 0) = 4 \)

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### 3. Problem c)
Minimize \( F(x, y) = 20.25 - y \) under the constraints:
- \( x \geq 0 \)
- \( y \geq 0 \)
- \( x \leq 3 \)
- \( -5x + 3y \leq 15 \)

**Constraints Interpretation:**
1. \( x \geq 0 \)
2. \( y \geq 0 \)
3. From \( x \leq 3 \) and rearranging \( -5x + 3y \leq 15 \) gives \( 3y \geq 5x - 15 \).

**Vertices of the feasible region:**
- Intersect the lines to find the vertices:
1. Intersection of \( x = 3 \) and \( 0 = 15 - 5(3) + 3y \) gives \( (3, 0) \)
2. Intersection when \( y = 0 \) gives \( x = 3 \) (on the boundary)
3. Check at \( (0, 0) \)

**Evaluating the objective function at the vertices:**
- \( F(0, 0) = 20.25 - 0 = 20.25 \)
- \( F(3, 0) = 20.25 - 0 = 20.25 \)

The line \( -5x + 3y = 15 \) allows us to find boundaries within the region constrained.

The actual maximum value will be dependent on the coordinates where \( y \) intervention happens.

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Feel free to adjust or ask for elaboration on any specific parts!