Lagrange Multiplier. Dealing with f(x,y) =xy^2

In summary, the conversation discusses the use of the Lagrange method to find the maximum and minimum of a function subject to a constraint. The question asks how to handle a function where the variables are multiplied together instead of being subtracted. It is explained that in this case, the partial derivatives of the function must be determined.
  • #1
King_Silver
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Given a question like this:
Findhe maximum and minimum of [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/eq0043M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/empty.gif subject to the constraint [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/eq0044M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/empty.gif.

I know that 5 = 2λx, -3 = 2λy.

However, if I am given a question where f(x,y) = xy2 how would I write this?
There is no sign separating the x and the y so I cannot simply presume what sign it is.

Would 1 = 2λx and 1 = 2λy be incorrect?
If so, why? and how is this sort of question dealt with when the x and y are being multiplied together instead of being subtracted? cheers
 
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  • #2
For the Lagrange method you look at the gradient of ##f - \lambda g##.
So all you have to do is determine the partial derivatives ##\partial f\over \partial x ## and ##\partial f\over \partial y ## . These aren't constants any more in this case.
 
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Related to Lagrange Multiplier. Dealing with f(x,y) =xy^2

1. What is a Lagrange Multiplier?

A Lagrange Multiplier is a mathematical tool used to find the maximum or minimum values of a function subject to one or more constraints. It involves using the gradient of the function and the constraints to find the critical points.

2. How is a Lagrange Multiplier used?

A Lagrange Multiplier is used to find the maximum or minimum values of a function subject to constraints. It involves setting up a system of equations using the gradient of the function and the constraints, and then solving for the critical points.

3. How do you solve for the Lagrange Multiplier?

To solve for the Lagrange Multiplier, you first set up a system of equations by taking the gradient of the function and the constraints. Then, you solve the system of equations to find the critical points. Finally, you plug the critical points back into the original function to find the maximum or minimum value.

4. What is the significance of Lagrange Multiplier in optimization?

Lagrange Multiplier is significant in optimization because it allows for the consideration of constraints when finding the maximum or minimum values of a function. Without it, the optimization process would be more limited and less accurate.

5. How is Lagrange Multiplier applied to f(x,y) = xy^2?

To apply Lagrange Multiplier to f(x,y) = xy^2, you would first set up the system of equations using the gradient of the function and any given constraints. Then, you would solve for the critical points and plug them back into the original function to find the maximum or minimum value. The Lagrange Multiplier would be the coefficient of the constraint in the system of equations.

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