Math Problem Statement
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.
Solution
The problem in the image asks for the minimum value of the function subject to the following constraints:
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)$$. Would you like more details or have any questions? --- Here are 5 related questions for deeper understanding: 1. How do boundary conditions affect the optimization of a function? 2. What is the significance of linear constraints in optimization problems? 3. Could the feasible region ever be empty for a similar set of constraints? 4. How would the solution change if the function to minimize was $$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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Inequalities
Optimization
Formulas
F(x, y) = x - 2y
Theorems
Feasibility region in optimization problems
Linear programming boundaries
Suitable Grade Level
Grade 10-12