Solve Highest/Lowest Points on Curve of Intersection with Lagrange Multipliers

In summary, the conversation discusses using Lagrange multipliers to find the highest and lowest points on the curve of intersection between an elliptic paraboloid and a right circular cylinder. The speaker has found the critical points, but is unsure how to determine if they are at maximum or minimum. They also question if there is a typo in the given equation for the elliptic paraboloid.
  • #1
Cherizzle
1
0
Hi There I would like help on a question about Lagrange multipliers.

Question: Consider the intersection of two surfaces: an elliptic paraboloid z=x^2 + 2*x + 4*y^2 and a right circular cylinder x^2 + y^2 = 1. Use Lagrange multipliers to find the highest and lowest points on the curve of the intersection.

What I have so far:
I managed to find my critical points using lagrange multipliers. But now I don't know how to describe whether my points are at maximum or minimum...
The points I found were: (2/3, -[tex]\sqrt{5}[/tex]/3) and (-[tex]\sqrt{2}[/tex]/6, 1)
 
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  • #2
Hi Cherizzle. Are you sure that's supposed to be z=x^2 + 2*x + 4*y^2 and not z=x^2 + 2*x*y + 4*y^2
 

FAQ: Solve Highest/Lowest Points on Curve of Intersection with Lagrange Multipliers

1. What is the concept of Lagrange multipliers?

Lagrange multipliers are a mathematical tool used to find the maximum or minimum value of a function subject to a set of constraints. This method involves creating a new function by combining the original function and the constraints, and then finding the critical points of this new function.

2. How is the method of Lagrange multipliers used to solve for the highest or lowest points on a curve of intersection?

To solve for the highest or lowest points on a curve of intersection using Lagrange multipliers, we first set up the system of equations by combining the equations of the curves and the constraint. Then, we use the method of Lagrange multipliers to find the critical points of this system. Finally, we substitute these critical points into the original equations to find the highest or lowest points on the curve of intersection.

3. Can Lagrange multipliers be used to find multiple critical points on a curve of intersection?

Yes, Lagrange multipliers can be used to find multiple critical points on a curve of intersection. This method can be applied to any number of equations and constraints, and it will provide all the possible critical points that satisfy the constraints.

4. What are the advantages of using Lagrange multipliers to solve for the highest or lowest points on a curve of intersection?

Using Lagrange multipliers to find the highest or lowest points on a curve of intersection is advantageous because it allows us to consider multiple equations and constraints at once, rather than solving each equation separately. This method also provides all the critical points that satisfy the constraints, making it more efficient and accurate.

5. Are there any limitations to using Lagrange multipliers to solve for the highest or lowest points on a curve of intersection?

One limitation of using Lagrange multipliers is that it may not always provide the global maximum or minimum point. It may only provide the local maximum or minimum point, depending on the constraints and the equations involved. Additionally, this method can become more complex and time-consuming as the number of equations and constraints increases.

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