Lagrange multiplier

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  • #1
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Homework Statement


Use the Lagrange Multiplier method to find the maximum and minimum values of x2 − 2xy + 7y2 on the ellipse x2 + 4y2 = 1.

Homework Equations


Lagrange multiplier method

The Attempt at a Solution


L(x,y,z,λ) = x2 − 2xy + 7y2 - λ(x2 + 4y2 - 1)
Find Lx, Ly, Lλ
Then, solve for x and y?
 
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Answers and Replies

  • #2
ideasrule
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Where did you get the L(x,y,z,λ)? You don't need the z in there because we're dealing with functions of x and y, not functions of x, y, and z.

Also, λ is a constant: you don't need to solve for Lλ. You don't need to solve for Lz either because there's no such variable as z. Just set Lx=0 and Ly=0 and solve. Don't forget to satisfy the initial constraint of x^2+4y^2=1!
 
  • #3
HallsofIvy
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Just an added note: some people are taught to find [itex]L_\lambda[/itex]. Because if we are trying to extremize F(x) subject to the constraint G(x)= 0, we look at [itex]L= F(x)+ \lambda G(x)[/itex], [itex]L_{\lamba}= G(x)= 0[/itex] is just the constraint again.
 
  • #4
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@ideasrule: Sorry, ignore the "z."
@HallsofIvy: Thanks. And yes, that's how I learned it.

Also, once I find the critical points, do I just plug them back into the original function I want to extremize to see if it's a max or min?
 

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