Lagrange Multipliers - basic which value?

In summary, the conversation discusses the use of summarizing content and the use of equations to solve for variables. The speaker also mentions the difficulty of identifying maximum and minimum values in Lagrange and the recommendation to use geometry instead.
  • #1
rocomath
1,755
1
(1) [tex]f(x,y,z)=x+2y[/tex]
(2) [tex]x+y+z=1[/tex]
(3) [tex]y^2+z^2=4[/tex]

[tex]1=\lambda[/tex]
[tex]2=\lambda+2y\mu[/tex]
[tex]0=\lambda+2z\mu[/tex]

[tex]u=\frac{1}{2y}[/tex]
[tex]y=\pm\sqrt2 \ \ \ z=\pm\sqrt2[/tex]

Plugging into equation 2 to solve for x.

How do I know to use either [tex]y=\sqrt 2 \ \mbox{or} \ y=-\sqrt2[/tex] ... similarly with my values for z.

edit: NVM, I'm an idiot :p I overlooked a step, which told me that [tex]z=-y[/tex]

... too late to delete?
 
Last edited:
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  • #2
see the geometry (It's a plane)

f(x,y) = ..

I was thinking about f_xx*f_yy - (f_xy)^2 thing,
but my teacher says it's very hard to *identify* max min in Lagrange; should use geometry
 
Last edited:

1. What is the concept of Lagrange Multipliers and how are they used?

Lagrange Multipliers are a mathematical tool used in multivariable calculus to find the maximum or minimum value of a function subject to a set of constraints. They are used to find optimal solutions in optimization problems where the constraints are in the form of equations.

2. How do Lagrange Multipliers work?

Lagrange Multipliers work by introducing a new variable, called the Lagrange multiplier, into the original function and the constraints. This new variable is multiplied by each constraint and then added to the original function. The resulting function is then solved for its critical points to find the optimal solution.

3. What is the purpose of using Lagrange Multipliers?

The purpose of using Lagrange Multipliers is to find the maximum or minimum value of a function subject to constraints. This allows for optimization in various fields such as economics, engineering, and physics.

4. What are the necessary conditions for using Lagrange Multipliers?

The necessary conditions for using Lagrange Multipliers include having a function to optimize, a set of constraints in the form of equations, and continuous first-order partial derivatives for all variables involved.

5. What are some real-life applications of Lagrange Multipliers?

Lagrange Multipliers have various real-life applications, including in economics for maximizing profit and minimizing costs, in physics for finding the path of least resistance, and in engineering for optimizing designs and systems.

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