Max Value f(r): Find Solution w/ Condition r^2=1

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Discussion Overview

The discussion revolves around finding the maximum value of the function f(r) = x² + 3y² + 2z² under the constraint r² = 1, where participants explore the application of the Lagrange multiplier method and clarify the relationship between the variables involved.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion regarding the notation f(r) since f is defined in terms of x, y, and z, not r.
  • There is a suggestion that r could represent either r = x² + y² + z² = 1 or r = x² + y² = 1, but this is not agreed upon.
  • One participant proposes using the Lagrange multiplier method to find the maximum value, indicating that the gradient should be normal to the constraint surface.
  • Another participant questions how to handle the gradient of f and the constraint function g, expressing concern about the implications for the value of lambda in the context of multiple variables.
  • There is a suggestion to adapt the formulas from the Lagrange multiplier method as presented in external resources to the specific case of three variables.

Areas of Agreement / Disagreement

Participants generally agree on the need to use the Lagrange multiplier method, but there is disagreement and confusion regarding the interpretation of the variables and the constraint, leading to unresolved questions about the correct approach.

Contextual Notes

Participants express uncertainty about the definitions of r and its relationship to x, y, and z, as well as the implications for the Lagrange multiplier method. There are unresolved mathematical steps related to the values of lambda and the gradients involved.

trelek2
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given:[tex] f(r)=x ^{2}+3y ^{2} +2z ^{2}[/tex]
The task was to calculate at the point (2,3,1): the grad of f, tangent plane, directional derivative in the direction (2,-1,0) but also to find the maximum value of f subject to the condition that.
[tex]r ^{2} =1[/tex]
I've done all except the last part, I have no idea what I am supposed to do here, and I don't really understand what they want.
Please explain.
 
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trelek2 said:
given:[tex] f(r)=x ^{2}+3y ^{2} +2z ^{2}[/tex]
The task was to calculate at the point (2,3,1): the grad of f, tangent plane, directional derivative in the direction (2,-1,0) but also to find the maximum value of f subject to the condition that.
[tex]r ^{2} =1[/tex]
I've done all except the last part, I have no idea what I am supposed to do here, and I don't really understand what they want.
Please explain.
Well, I don't either because there is not "r" given. If I had to guess it would be either [itex]r= x^2+ y^2+ z^2= 1[/itex], although I would be inclined to use "[itex]\rho[/itex]", or [itex]r= x^2+ y^2= 1[/itex].
 
Why do you write "f(r)" when f is a function of x,y,z and can not be written as a function of the radius?
To find the maximum you should probably use the Langrange multiplier method (find points where gradient is normal to the set on which f should be optimized).
Link: http://en.wikipedia.org/wiki/Lagrange_multipliers"
 
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It is what i have in the exercise...
I also have problems distinguishing r and the x+y+z stuff. How would you treat it?
So supposing r^2=1=x^2+y^2+z^2, should I then take the gradient of f(r) at the given point to find the value of the langrange multiplayer?
 
Last edited:
Apply the Langrange multiplier method (as in the wikipedia article or maybe in your notes/textbook) to the function f(x,y,z)=x^2+3y^2+2z^2 and the constraint g(x,y,z)=x^2+y^2+z^2=1. You have x,y,z instead of just x,y as in the wikipedia article, but you should be able to adapt the formulas easily.
 
I'm still really confused how to do this. Since I get the gradient of f and the constraint function g in terms of (2x+2xlambda, 6y+2ylambda,4z+2zlambda) It seems to imply that lambda has to be 3 different values at the same time as the variables get reduced. To keep the variables I can take as stated in the exercise at the point f(2,3,1) but does that make any sense?
 

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