Optimizing Constraints: Finding Min/Max Values Of C=x+y

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Hey guys! Let's dive into a fun math problem. We're going to explore how to find the minimum and maximum values of an objective function, given a set of constraints. This is a classic optimization problem, and it's super useful in all sorts of real-world scenarios, from business to engineering. So, buckle up and let's get started! Our goal is to figure out where the objective function, which is c = x + y, hits its lowest and highest points. But, we can't just pick any x and y values we want. We have some rules, or constraints, that limit our choices. These constraints are essentially inequalities that define a feasible region, and any solution to this problem must exist in this region. Let's break down each constraint and then figure out how to find our minimum and maximum values.

Understanding the Constraints

First off, our objective function c = x + y is a linear equation that we want to either minimize or maximize. Think of it like a hill that we want to climb up or down. The constraints are what define the terrain of our hill. Let's unpack the constraints individually:

  1. y \leq 5: This constraint tells us that the value of y must be less than or equal to 5. Graphically, this means all the points on or below the horizontal line y = 5 are allowed. This line acts as a hard upper boundary for our solutions.
  2. y \geq 0: This constraint means that the value of y must be greater than or equal to 0. This is a common-sense constraint, as we're often dealing with quantities that can't be negative. This tells us that we can only consider points on or above the x-axis.
  3. y - 2x \geq 0: This is the more interesting constraint. It can be rewritten as y \geq 2x. This tells us that the value of y must be greater than or equal to twice the value of x. Graphically, this is a line through the origin (0,0), with a slope of 2. Everything above this line is included in the feasible region. This constraint essentially carves out a wedge-shaped region in our graph.

So, the feasible region is where all these constraints overlap. To visualize the feasible region, plot each constraint on a graph. The area where all three conditions are met is our feasible region. It's like the only part of the map where our treasure (the minimum and maximum values) could be hidden. The feasible region is going to be a closed polygon because all the constraints are linear.

Graphing the Constraints and Identifying the Feasible Region

Okay, let's get our hands dirty and start plotting this stuff! Grab some graph paper (or use a digital graphing tool – they're awesome) and follow along. First, draw your axes: the x-axis (horizontal) and the y-axis (vertical). Label them clearly. Now, let's plot each constraint.

  1. y \leq 5: Draw a horizontal line at y = 5. Since y must be less than or equal to 5, shade the area below this line. This area represents all the points where this constraint is satisfied.
  2. y \geq 0: The x-axis itself is y = 0. Shade the area above the x-axis (including the x-axis itself). This ensures that y is greater than or equal to zero.
  3. y - 2x \geq 0 or y \geq 2x: This one's a bit different, but no worries, we got this. The equation y = 2x is a straight line that goes through the origin (0,0). To plot it, find at least two points that satisfy the equation. For instance, when x = 1, y = 2; when x = 2, y = 4. Plot these points and draw the line. Because the constraint is y \geq 2x, shade the area above this line. This represents all the points where y is greater than or equal to 2x.

Now, here's the key: The feasible region is the area where all three shaded regions overlap. It's the only area on the graph where all constraints are satisfied simultaneously. In our case, the feasible region will be a closed polygon. It will be a region bounded by the lines y = 5, y = 0, and y = 2x. This region represents all the possible combinations of x and y that meet our criteria. The most important part of this feasible region is the vertices, or the corner points, where the boundary lines intersect. These vertices hold the key to finding the minimum and maximum values of the objective function. Understanding and correctly identifying the feasible region is crucial to solving the optimization problem. If we get this part wrong, then the whole process is doomed from the start. Spend extra time ensuring you have accurately represented each constraint and the region.

Finding the Vertices of the Feasible Region

Alright, now that we have our feasible region, the next step is to identify the vertices. Vertices are the corner points of the feasible region. These are the points where the boundary lines of our constraints intersect. Why are they important? Because the minimum and maximum values of our objective function always occur at one of these vertices (or along an edge, which is also determined by the vertices). To find the vertices, we need to solve the system of equations formed by the intersecting lines.

Let's figure out the vertices of our feasible region: We have the following intersections:

  1. Intersection of y = 5 and y = 2x: Substitute y = 5 into y = 2x to get 5 = 2x. Solving for x, we find x = 2.5. So, one vertex is (2.5, 5).
  2. Intersection of y = 0 and y = 2x: Substituting y = 0 into y = 2x, we get 0 = 2x, which means x = 0. So, another vertex is (0, 0).
  3. Intersection of y = 5 and y = 0: This is impossible and does not form a vertex, as it is a contradiction. The correct intersection should be y = 0 and y = 2x gives x = 0. So, another vertex is (0,0) and y = 5 and x = 0 is not possible, so it is (0, 5) and this gives another vertex. Notice in this case, we have a triangle-shaped feasible region, formed by the three vertices (0, 0), (2.5, 5), and (0, 5). These are our vertices, the key to unlocking our minimum and maximum values. Make sure you accurately find these points. Incorrect vertices will lead to an incorrect answer!

Evaluating the Objective Function at the Vertices

Now comes the fun part! We have our vertices, and we're ready to see what happens to our objective function, c = x + y, at each of these points. Remember, the objective function represents the value we want to minimize or maximize. By plugging in the x and y values of our vertices into this function, we can determine where the minimum and maximum values occur.

Let's evaluate c = x + y at each vertex:

  1. At the vertex (0, 0): c = 0 + 0 = 0.
  2. At the vertex (2.5, 5): c = 2.5 + 5 = 7.5.
  3. At the vertex (0, 5): c = 0 + 5 = 5.

By evaluating the objective function at each vertex, we've found the values of c corresponding to each corner point. Now, compare these values to identify the minimum and maximum. So, at (0,0), c = 0; at (2.5, 5), c = 7.5; and at (0, 5), c = 5.

Determining the Minimum and Maximum Values

Drumroll, please! We're at the final step. Now it's time to look at the values we calculated for our objective function at each vertex and decide which one is the smallest (the minimum) and which one is the largest (the maximum).

Looking at our results from the previous step: We have 0, 7.5, and 5. Simple, right?

  • Minimum Value: The smallest value is 0. This occurs at the vertex (0, 0). Therefore, the minimum value of the objective function c = x + y is 0.
  • Maximum Value: The largest value is 7.5. This occurs at the vertex (2.5, 5). Therefore, the maximum value of the objective function c = x + y is 7.5.

And there you have it! We've successfully found both the minimum and maximum values of our objective function, subject to the given constraints. We went from a set of inequalities to a concrete answer. See, optimization problems aren't so scary, are they? Understanding the constraints, graphing them correctly, finding the feasible region, identifying the vertices, and then plugging them into the objective function is the whole process, and we just nailed it! You can use this methodology to tackle many other optimization problems.

Conclusion

So, we've covered a lot of ground today. We started with a set of constraints, visualized them graphically, identified the feasible region, found the vertices, and finally, evaluated our objective function to find its minimum and maximum values. Remember, optimization problems like these are super common in areas like economics, business, and engineering. Knowing how to solve them can be a valuable skill. Keep practicing, and you'll become a pro in no time!

Hopefully, this explanation has been helpful. Keep in mind that the techniques we discussed here can be extended to more complex scenarios, with more variables and constraints. But the core principles remain the same. So, next time you encounter an optimization problem, you'll be ready to tackle it head-on. Keep exploring, keep learning, and keep those math skills sharp, guys! If you have any questions, feel free to ask. Happy optimizing!