Max/Min Values for f(x,y,z): Lagrange Multipliers

In summary, Lagrange multipliers are a powerful tool for finding the extreme values of a multivariable function subject to a constraint. To use them, the function and constraint equations are set up and a new function, the Lagrangian, is created. This method can be used for any continuous function and constraint, but it has limitations such as only finding local extrema and being computationally expensive for large functions. However, it has many real-world applications in various fields.
  • #1
PhysicsMajor
15
0
Greetings all,

Find the max and min values of
f(x,y.z)=3x-y-3z
subject to
x+y-z=0, x^(2)+2z^(2)=1

can anybody help me get this problem started.

thanks
 
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  • #2
Just write the Lagrange function of the problem:

[tex] F(x,y,z)=3x-y-3z+\lambda_1(x+y-z)+\lambda_2(x^{2}+2z^{2}-1)[/tex]

Then find the partial derivatives (they have to be 0), and solve the system of equations.
 
  • #3
You'll find the critical points,though...You can't be sure which is a mex & which is a min...

Daniel.
 

Related to Max/Min Values for f(x,y,z): Lagrange Multipliers

1. What is the significance of finding the max/min values for a function using Lagrange multipliers?

Lagrange multipliers are a powerful mathematical tool used to find the extreme values (such as max/min) of a multivariable function subject to a constraint. This is extremely useful in optimization problems, as it allows us to find the most efficient solution while satisfying certain constraints.

2. How do you use Lagrange multipliers to find the max/min values for a function?

To use Lagrange multipliers, we first set up the function we want to optimize and the constraint equation. Then, we use the method of Lagrange multipliers to create a new function, known as the Lagrangian, by adding a new variable (lambda) to the original function. We then take the partial derivatives of the Lagrangian with respect to all variables and set them equal to 0. Solving these equations will give us the values of x, y, and z that correspond to the max/min values of the original function.

3. Can Lagrange multipliers be used for any function and constraint?

Yes, Lagrange multipliers can be used for any continuous function and constraint. However, they are most commonly used for functions with multiple variables and linear constraints.

4. What are the limitations of using Lagrange multipliers?

One limitation of using Lagrange multipliers is that it can only find local max/min values, not global max/min values. Additionally, this method can become computationally expensive for functions with a large number of variables.

5. Are there any real-world applications of Lagrange multipliers?

Yes, Lagrange multipliers have many real-world applications in fields such as economics, physics, engineering, and chemistry. For example, they can be used to optimize production processes, minimize costs in engineering designs, and find the most stable configuration of molecules in chemistry.

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