Solving Lagrange Problem Near (9,12,5)

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In summary, to solve the Lagrange problem near (9,12,5), one must determine the objective function and constraints, use the Lagrange multiplier method to find the critical points, and evaluate the solution to check for constraint satisfaction. The significance of this problem lies in its real-world applications and the fact that it can have multiple solutions. Unlike other optimization problems, the Lagrange problem near (9,12,5) uses the Lagrange multiplier method, and its practical applications range from production planning to physics.
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navalstudent
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Hi, I am supposed to find the point on the cone z^2=x^2+y^2 which is closest to the point(9,12,5).

here is my work:

http://img27.imageshack.us/my.php?image=lagrange001.jpg

Is it correct so far?

If it is: I get stuck when trying to solve the equations z^2=x^2+y^2, x=9/12*y, and -2xz+9z+5x=0. Can someone please give me some advice?
 
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  • #2
It looks to me like you are on the right track. Now use the second equation to eliminate x or y. It will work. Just keep going.
 

1. How do you approach solving the Lagrange problem near (9,12,5)?

To solve the Lagrange problem near (9,12,5), I would first determine the objective function and the constraints. Then, I would use the Lagrange multiplier method to find the critical points and determine which one is a minimum or maximum. Finally, I would evaluate the solution and check if it satisfies the constraints.

2. What is the significance of the Lagrange problem near (9,12,5)?

The Lagrange problem near (9,12,5) is significant because it is a mathematical optimization problem that involves finding the minimum or maximum value of a function subject to constraints. It has many real-world applications in fields such as economics, engineering, and physics.

3. Can the Lagrange problem near (9,12,5) have multiple solutions?

Yes, the Lagrange problem near (9,12,5) can have multiple solutions. This is because the objective function and constraints may have multiple critical points that satisfy the necessary conditions for optimization. In some cases, these solutions may be equivalent, while in others, they may yield different optimal values.

4. How does the Lagrange problem near (9,12,5) differ from other optimization problems?

The Lagrange problem near (9,12,5) differs from other optimization problems in that it uses the Lagrange multiplier method, which involves adding additional variables (the multipliers) to the objective function to incorporate the constraints. This allows the problem to be solved using techniques from calculus, rather than linear algebra or other methods used in other optimization problems.

5. What are some practical applications of solving the Lagrange problem near (9,12,5)?

The Lagrange problem near (9,12,5) has many practical applications, such as in production planning, resource allocation, and portfolio optimization. It can also be used in machine learning and data analysis to find the best fitting model or parameters for a given dataset. Additionally, it has applications in physics, such as in finding the path of least resistance or least action in a system.

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