Maximum and Minimum : Langrange multiplier problem

In summary, the conversation is about finding the maximum and minimum of a function over a closed and bounded set using the Langrange multiplier method. The first step is to set the first order partial derivatives to 0 to find the critical points in the interior. Then, the points that satisfy the equation x2/4 + y2/16 = 1 are checked for potential extremums using the Langrange multiplier method. The conversation also discusses the process of solving for lambda and how to get the solutions y=-x-1 and y=4x.
  • #1
michonamona
122
0

Homework Statement



Find the maximum and minimum of the function f over the closed and bounded set S. Use langrange multiplier method to find the values of the boundary points.

Homework Equations



f(x,y) = (1+x+y)2

S = {(x,y) : x2/4 + y2/16 <= 1}


The Attempt at a Solution




First, I set their first order partial derivatives to 0 to get the following

fx(x,y) = 2(1+x+y)=0
fy(x,y) = 2(1+x+y)=0.

It's obvious that I'm not going to be able to find a unique value for my critical points with these two equations, thus, I conclude that there are infinitely many critical points in the interior (I also don't understand the intuition behind this conclusion).

Next, we check the points that satisfy x2/4 + y2/16 = 1, to see if these are potential extremums. I will use the langrange multiplier method.

1. 2(1+x+y) = [tex]\lambda[/tex]x/2
2. 2(1+x+y) = [tex]\lambda[/tex]y/8
3. x2/4 + y2/16 = 1

Then the textbook says that solving for [tex]\lambda[/tex] will yield y=-x-1 or y=4x. I know how to get y=4x, but where did y=-x-1 come from? How were they able to derive it from these three equations?

Thank you very much for your help,

M
 
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  • #2
Your first case is for lambda different than zero, so you can identify the first two equations and then divide by lambda.
If lambda equals zero, then you get 1+x+y=0, which is the second solution.
 

1. What is the Langrange multiplier problem?

The Langrange multiplier problem is a mathematical optimization problem that involves finding the maximum or minimum value of a function subject to certain constraints. It is named after Joseph-Louis Lagrange, who first described the method in the late 18th century.

2. How does the Langrange multiplier method work?

The Langrange multiplier method involves setting up a system of equations using the original function and the constraints. The solution to this system of equations will give the maximum or minimum value of the function, taking into account the constraints. The method works by finding the points where the gradient of the function is parallel to the gradient of the constraint functions.

3. When is the Langrange multiplier method used?

The Langrange multiplier method is used when optimizing a function that is subject to one or more constraints. This can occur in various fields such as economics, physics, and engineering, where certain restrictions or limitations must be considered in order to find the optimal solution.

4. What are some common applications of the Langrange multiplier method?

The Langrange multiplier method is commonly used in economics to solve optimization problems involving production, consumption, and pricing. It is also used in physics to optimize systems under given constraints, such as finding the path of least resistance in a circuit. In addition, it is used in engineering to optimize designs while considering various constraints and limitations.

5. Are there any limitations to the Langrange multiplier method?

While the Langrange multiplier method is a powerful tool for solving optimization problems, it does have some limitations. It is only applicable for finding the maximum or minimum value of a function, and cannot be used for other types of optimization problems. Additionally, the method can become computationally intensive for problems with a large number of constraints.

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