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Optimisation using constraints

  • #1

Homework Statement



Consider the intersection of two surfaces: an elliptic paraboloid
z = x2 + 2x + 4y2 and a right circular cylinder x2 + y2 = 1. Use Lagrange multipliers to find
the highest and lowest points on the curve of intersection

The Attempt at a Solution



L = x^2 + 2x + 4y^2 - λ(x^2 + y^2 - 1)
Lx = 2x + 2 - 2λx = 0
Ly = 8y - 2λy = 0

Rearranging gives the bizzare result of λ = 4. Aren't I supposed to be able to eliminate the lagrange multipliers by it giving me λ in terms of y? It's giving me a value instead.

I ran with this anyway, and got the answer of x = 1/3, y = +-0.94, but I don't think this is correct. I suspect I'm doing something wrong algebraically? Any help hugely appreciated :)
 

Answers and Replies

  • #2
hotvette
Homework Helper
996
5
I suspect I'm doing something wrong algebraically? Any help hugely appreciated :)
You need to be careful with the last equation. I believe you divided by y, but that's valid only if y isn't zero. You need to explore both possibilities. By the way, you forgot an equation. What about Lλ?

What makes you think you have the wrong answer?
 
Last edited:
  • #3
Dick
Science Advisor
Homework Helper
26,258
618
If you don't trust the method try it directly as well. Use the constraint to eliminate y^2 in z. Now look at the extrema for x in [-1,1]. You do need to consider the y=0 case as well.
 

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