Solve Lagrange Multipliers: Find Max/Min f(x,y)

In summary, the problem involves using Lagrange multipliers to find the maximum and minimum values of a function subject to a given constraint. The critical points can be found in the interior region by setting the partial derivatives of the function and constraint equal to each other. However, since the partial derivatives are constants, there may not be any critical points inside the given constraint. Therefore, the maximum and minimum values must be found on the boundary of the constraint, which can be done by substituting the relationship between the partial derivatives into the constraint equation and solving for the possibilities of x and y.
  • #1
mirandasatterley
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Homework Statement



Use lagrange multipliers to find the maximum and minimum values of f subject to the given constraint, if such values exist.
f(x,y) = x+3y, x2+y2≤2


Homework Equations



grad f = λ grad g

The Attempt at a Solution



to find critical points in the interior region,
The partial derivative of f(x,y) with respect to x is 1.
The partial derivative of f(x,y) with respect to y is 3.

g(x,y) = the constraint = x2+y2≤2, to find critical points on the boundary x2+y2=2
The partial derivative of g(x,y) with respect to x is 2x.
The partial derivative of g(x,y) with respect to y is 2y.

And normally I would set:
1 = λ 2x and 3 = λ 2y
λ = 1/2x and λ = 3/2y
so then I would set the equations equal to each other and solve the equation for x and y.

What I'm wondering though, Is whether there are actually minimum or maximum values since the partial derivatives of f(x,y) are constants?
 
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  • #2
grad f is nonzero everywhere. So there are no critical points (local maxs or mins) inside the circle x^2+y^2=2. So the max and/or min must be ON the circle. You've found 1/(2x)=3/(2y), substitute that into x^2+y^2=2 and find the possibilities for x and y.
 

1. What is the Lagrange multiplier method?

The Lagrange multiplier method is a mathematical optimization technique used to find the maximum or minimum value of a function subject to one or more constraints.

2. How do I use Lagrange multipliers to find the maximum or minimum value of a function?

To use Lagrange multipliers, you must first set up the Lagrangian function by combining the original objective function and the constraints. Then, you can solve for the critical points of the Lagrangian function to find the maximum or minimum value.

3. What are the necessary conditions for using Lagrange multipliers?

The necessary conditions for using Lagrange multipliers are that the objective function and constraints must be smooth and the constraints must be independent of each other.

4. Can Lagrange multipliers be used for non-linear programming problems?

Yes, Lagrange multipliers can be used for non-linear programming problems. However, in some cases, it may be more efficient to use other optimization techniques.

5. Are there any limitations to using Lagrange multipliers?

One limitation of using Lagrange multipliers is that it can only find local extrema, not global extrema. Additionally, the method may become computationally intensive for problems with a large number of constraints.

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