Lagrange Multipliers (and finding extrema of a function with two restraints)

In summary, Lagrange Multipliers are a mathematical tool used to optimize a function subject to constraints. They are important in real-world applications and can be used for functions with any number of constraints. Some common mistakes when using Lagrange Multipliers include forgetting to include all constraints and not verifying the critical point. To use Lagrange Multipliers with two constraints, we set up a system of equations and solve for the values of the variables that maximize or minimize the function while satisfying the constraints.
  • #1
Black Orpheus
23
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I need to find the extrema of f(x,y,z)=x+y+z subject to the restraints of x^2 - y^2 = 1 and 2x+z = 1. So the gradient of f equals (1,1,1) =
lambda1(2x,-2y,0) + lambda2(2,0,1). Solving for the lambdas I found that lambda1 = -1/(2x) = -1/(2y), or x=y. But this isn't possible if x^2 - y^2 = 1. Does this mean that there are no absolute max or min, or am I doing something wrong?
 
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  • #2
It may mean that the max or min is on the boundary of the set.
 
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What are Lagrange Multipliers?

Lagrange Multipliers are a mathematical tool used to find the extrema of a function subject to one or more constraints. They allow us to optimize a function with certain restrictions, such as a limited budget or physical constraints.

Why are Lagrange Multipliers important?

Lagrange Multipliers are important because they provide a systematic and efficient way to find the extrema of a function while considering constraints. This is useful in many real-world applications, such as economics, engineering, and physics.

How do you use Lagrange Multipliers to find extrema of a function with two constraints?

To use Lagrange Multipliers to find extrema of a function with two constraints, we first write out the function and the two constraints as equations. Then, we set up the Lagrange Multiplier equation by introducing a new variable, lambda, and taking the partial derivatives of the function and constraints with respect to all variables. Finally, we solve the system of equations to find the values of the variables that maximize or minimize the function while satisfying the constraints.

What are some common mistakes when using Lagrange Multipliers?

Some common mistakes when using Lagrange Multipliers include forgetting to include all constraints in the Lagrange Multiplier equation, not considering the possibility of multiple critical points, and not verifying that the critical point is a maximum or minimum by using the second derivative test.

Can Lagrange Multipliers be used for functions with more than two constraints?

Yes, Lagrange Multipliers can be used for functions with any number of constraints. The process is the same as when using two constraints, but there will be more equations to solve for the values of the variables.

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