Lagrange multipliers.

  • Thread starter squenshl
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  • #1
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Homework Statement


Minimise = x2 + y2 subject to C(x,y) = 4x2 + 3y2 = 12.


Homework Equations





The Attempt at a Solution


I let h(x,y) = x2 + y2 + [tex]\lambda[/tex](4x2 + 3y2 - 12).
I got hx = 2x + 8[tex]\lambda[/tex]x = 0, hy = 2y + 6[tex]\lambda[/tex]y = 0, but here I get 2 values of [tex]\lambda[/tex], [tex]\lambda[/tex] = -1/4 & -1/3. So I don't get a min/max value.
 

Answers and Replies

  • #2
fzero
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When you solve each equation for [tex]\lambda[/tex], you're making an assumption about the values of x and/or y. Perhaps one of these assumptions is wrong.
 
  • #3
Dick
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How about, for example, lambda=(-1/4) and y=0. Doesn't that satisfy both equations?
 
  • #4
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So a min/max point is (0,0). How do I get the other 3.
 
  • #5
vela
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(0,0) doesn't satisfy the constraint. Try again.
 
  • #6
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I think I got it. 4 points: (0,2),(0,-2),([tex]\sqrt{3}[/tex],0),(-[tex]\sqrt{3}[/tex],0). When it asks to minimise do I just find the 2 min values.
 
  • #7
vela
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Yup.
 
  • #8
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Cheers.
I have anther question.
How do I find the point(s) on the surface z2 + xy = 1 that lie closest to the origin.
Is it let h(x,y,z) = x2 + y2 + z2 + [tex]\lambda[/tex](z2 + xy - 1) then find the partial derivatives & equate to 0 & the rest.
 
  • #9
vela
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Yup.
 
  • #10
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I get [tex]\lambda[/tex] = 0 so the point is (0,0,0) which is the origin but that can't be right?
 
  • #11
vela
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No, that point doesn't satisfy the constraint, so the multiplier can't be 0.

Look at the z equation. Assume z isn't 0. What does λ have to be? Then use that value to solve for x and y. Then solve for z.

Now assume z=0. To satisfy the constraint, neither x nor y can vanish. For what values of λ will the x and y equations have non-zero solutions?
 
  • #12
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Oops. I got my hz wrong. [tex]\lambda[/tex] = -1. So y = 2x & x = 2y. So we get z2 + 2x3 = 1 therefore z = [tex]\pm[/tex] (1-2x2)1/2. I get (0,0,1) & (0,0 -1). I actually get 4 more points making 6 in total.
 
Last edited:
  • #13
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I think this is wrong.
 
  • #14
vela
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Why?
 

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