Use lagrange multipliers to find the shortest distance

In summary, the shortest distance between a point on the elliptic paraboloid z=x^2+y^2 and the constraint x^2+y^2-z=0 is found using Lagrange multipliers. After differentiating the equation with respect to lambda and setting it equal to zero, a mistake is identified in the third step which is corrected. This helps in finding the correct solution for lambda and solving the resulting equation.
  • #1
anubis01
149
1

Homework Statement


Use lagrange multipliers to find the shortest distance between a point on the elliptic paraboloid z=x^2 +y^2


Homework Equations





The Attempt at a Solution


http://img716.imageshack.us/img716/7272/cci1902201000000.jpg [Broken]

I'm not that good with using the equation editor, so I scanned my work.

I'm stuck on the last part where I'm trying to factor the equation to find a solution for [tex]\lambda[/tex], I can't seem to find a solution that would make the equation zero, which is what i need in order to do the long division to factor that equation.
 
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  • #2
Your constraint is x^2+y^2-z=0. You differentiate lambda*(x^2+y^2-z). So you have a mistake on step 3. The d/dz equation should be 2*(z-1/2)=(-lambda). That may be why you are having a hard time with the resulting equation.
 
  • #3
ah, now the equation makes much more sense, thanks for the help.
 

What is the purpose of using Lagrange multipliers to find the shortest distance?

The purpose of using Lagrange multipliers is to optimize a function subject to a set of constraints. In the case of finding the shortest distance, Lagrange multipliers allow us to find the minimum distance between two points while considering any additional constraints that may exist.

How does the Lagrange multiplier method work?

The Lagrange multiplier method involves finding the gradient of the objective function and the constraints, and then setting them equal to each other. This creates a system of equations that can be solved to find the optimal solution.

What are the benefits of using Lagrange multipliers to find the shortest distance?

Using Lagrange multipliers allows for a more efficient and accurate solution to finding the shortest distance. It takes into account any constraints that may exist and provides a rigorous mathematical approach to finding the optimal solution.

Can Lagrange multipliers be used for finding the shortest distance in higher dimensions?

Yes, Lagrange multipliers can be used for finding the shortest distance in higher dimensions. The method remains the same, but the equations become more complex as the number of dimensions increases.

Are there any limitations to using Lagrange multipliers for finding the shortest distance?

One limitation of using Lagrange multipliers is that it may not always provide the global minimum distance. It is possible for the method to find a local minimum instead. Additionally, the calculations can become very complex and time-consuming in higher dimensions.

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