https://file.aiquickdraw.com/mathbot/file/1729698754348v44e2sou.jpg

The problem in the image asks for the minimum value of the function \( F(x, y) = x - 2y \) subject to the following constraints:

\[
\begin{cases}
0 \leq y \leq 5 \\
x \geq 0 \\
x + y - 2 \geq 0 \\
x - y - 2 \leq 0
\end{cases}
\]

### Steps to solve:

1. **Analyze the constraints:**
   - \( 0 \leq y \leq 5 \) ensures \( y \) is within this interval.
   - \( x \geq 0 \) means \( x \) is non-negative.
   - \( x + y - 2 \geq 0 \Rightarrow x \geq 2 - y \).
   - \( x - y - 2 \leq 0 \Rightarrow x \leq y + 2 \).

2. **Determine the region of feasibility:**
   Combining all constraints, the feasible region is determined by:
   \[
   \max(0, 2 - y) \leq x \leq y + 2.
   \]
   This restricts \( x \) to a certain range depending on \( y \).

3. **Find the minimum value of \( F(x, y) = x - 2y \):**
   We need to minimize \( x - 2y \) over the feasible region. The extreme values typically occur at the boundary of the constraints.

   - For \( y = 0 \):
     - \( x \geq 2 - 0 = 2 \) and \( x \leq 0 + 2 = 2 \), so \( x = 2 \).
     - \( F(2, 0) = 2 - 2(0) = 2 \).

   - For \( y = 5 \):
     - \( x \geq 2 - 5 = -3 \) (but since \( x \geq 0 \), we take \( x = 0 \)).
     - \( x \leq 5 + 2 = 7 \), so we evaluate \( F(0, 5) = 0 - 2(5) = -10 \).

4. **Conclusion:**
   The minimum value of \( F(x, y) = x - 2y \) is \( \boxed{-10} \), which occurs at \( (x, y) = (0, 5) \).

Would you like more details or have any questions?

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Here are 5 related questions for deeper understanding:
1. How do boundary conditions affect the optimization of a function?
2. What is the significance of linear constraints in optimization problems?
3. Could the feasible region ever be empty for a similar set of constraints?
4. How would the solution change if the function to minimize was \( F(x, y) = 2x - y \)?
5. What is the difference between solving a linear optimization problem and a nonlinear one?

**Tip:** When optimizing a function with constraints, always check the boundaries first since extreme values often occur there.

Tìm giá trị nhỏ nhất của biểu thức F(x; y) = x - 2y với điều kiện 0 ≤ y ≤ 5, x ≥ 0, x + y - 2 ≥ 0, x - y - 2 ≤ 0.

Learn how to find the minimum value of F(x, y) = x - 2y given constraints such as 0 ≤ y ≤ 5 and several linear inequalities. This problem focuses on optimization methods and linear algebra concepts suitable for students in Grades 10-12.

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