Understanding Lagrange Multipliers: Solving for Max and Min Values

Click For Summary

Discussion Overview

The discussion revolves around the application of Lagrange multipliers to find the maximum and minimum values of the function f(x,y,z) = 3x + 2y + z, subject to the constraint g(x,y,z) = x² + y² + z² = 1. Participants seek clarification on the method and steps involved in solving this problem, particularly focusing on the initial steps of the Lagrange multiplier technique.

Discussion Character

  • Homework-related
  • Technical explanation

Main Points Raised

  • One participant expresses confusion about how to proceed after stating the function and constraint, asking for help with Lagrange multipliers.
  • Another participant provides a summary of the Lagrange multipliers method, outlining the steps to find maximum and minimum values, but does not confirm whether the original poster understands these steps.
  • A participant reiterates their lack of understanding regarding the steps, specifically mentioning difficulty with the concept of partial derivatives and the gradient.
  • A further explanation is provided about the gradient and its relation to the Lagrange multiplier method, detailing how to set up the vector equation.
  • There is an offer to explain the underlying rationale behind the method, indicating that it may be complex to grasp.

Areas of Agreement / Disagreement

Participants generally express confusion and seek clarification on the method, indicating a lack of consensus on understanding the initial steps of applying Lagrange multipliers.

Contextual Notes

Participants have not reached a resolution regarding the understanding of the method, and there are indications of varying levels of familiarity with the mathematical concepts involved, such as gradients and partial derivatives.

Dx
Find max and min value…f(x,y,z) =3x+2y+z; x2 + y2+z2 = 1

If g(x,y,z) = x2 + y2+z2 = 1 then what do I do next?

I need help to further solve for this please? I am horrible at math and don't understand lagrange multipiers so can anyone better explain it to me and help me solve for difficult problem.

Dx
 
Physics news on Phys.org
Well, my text has the following box for lagrange multipliers (interesting theorems / procedures are placed in boxes in the text):

To find the maximum and minimum values of f(x, y, z) subject to the constraint g(x, y, z) = k (assuming these extreme values exist):

(a) Find all values of x, y, z, and λ such that:

[nab]f(x, y, z) = λ [nab]g(x, y, z)

and g(x, y, z) = k

(b) Evaluate f at all the points (x, y, z) that arise from step (a). The largest of these values is the maximum value of f; the smallest is the minimum value of f.

I imagine your text has something similar. Do you understand how to start step (a)?
 
Originally posted by Hurkyl
Well, my text has the following box for lagrange multipliers (interesting theorems / procedures are placed in boxes in the text):



I imagine your text has something similar. Do you understand how to start step (a)?

No, I don't understand this what-so-ever. I see one example in my book that wants to find the points of a rectangular hyperbola and its using partial Dx but I don't know how to perform step a could you please help me.

Dx
 
∇f(x, y, z) means the gradient of f with respect to its three variables. The gradient is defined as the row vector:

&nabla;f(x, y, z) = < &part;f(x, y, z)/&part;x, &part;f(x, y, z)/&part;y, &part;f(x, y, z)/&part;z >
I.E. the first coordinate is the partial derivative with respect to (WRT) the first variable, the second coordinate is the partial derivative WRT the second variable, and so on for as many variabes as the function has.


The equation in step (a) is, then, a vector equation, which we solve by setting corresponding coordinates equal:

&nabla;f(x, y, z) = &lambda; &nabla;g(x, y, z)

is

< &part;f(x, y, z)/&part;x, &part;f(x, y, z)/&part;y, &part;f(x, y, z)/&part;z > = &lambda; < &part;g(x, y, z)/&part;x, &part;g(x, y, z)/&part;y, &part;g(x, y, z)/&part;z >

is

&part;f(x, y, z)/&part;x = &lambda; &part;g(x, y, z)/&part;x
&part;f(x, y, z)/&part;y = &lambda; &part;g(x, y, z)/&part;y
&part;f(x, y, z)/&part;z = &lambda; &part;g(x, y, z)/&part;z



(BTW, if you also want me to explain the "why" behind this method, ask and I'll do so... but the "why" may be quite difficult to understand)
 

Similar threads

Undergrad The Euler-Lagrange equation and the Beltrami identity
  • · Replies 1 ·
Replies
1
Views
3K
MHB Solve Lagrange Multipliers Problem w/ e^(9x) - Find (a,b)
  • · Replies 2 ·
Replies
2
Views
1K
MHB Solve Lagrange Multipliers Problem with x-4y=1 Constraint
  • · Replies 1 ·
Replies
1
Views
2K
MHB Solving Lagrange Multipliers: Find Extrema of Distance from (1,2,3) to Sphere
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Lagrange multipliers: How do I know if its a max of min
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Need help understanding Lagrange multipliers at a more fundamental level.
  • · Replies 3 ·
Replies
3
Views
3K
Lagrange multipliers understanding
  • · Replies 8 ·
Replies
8
Views
2K
MHB Holly's questions at Yahoo Answers regarding Lagrange multipliers
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Can someone help me solve these problems?
  • · Replies 1 ·
Replies
1
Views
1K