Lagrange Multipliers

[Astronouth7303]

Ok, I’m having a problem understanding a question about Lagrange Multipliers.

Given the function f(x, y) = 16 − x2 − y2 and the constraint (x − 1)2 + (y − 1)2 = 4, find the maximum and minimum.

I’ve reduced it to the system:

1. -2x
   = λ∙2x
2. -2y
   = λ∙2y
3. (x
   − 1)2 + (y − 1)2 = 4

There are a few ways to solve this:

• Assume λ = -1, which gets you nowhere
• Assume x
  = 0, for which y = 1 ± √3
• Assume y
  = 0, for which x = 1 ± √3

I know that the real answer is when x = y, but I can’t get that going forwards.

Anyone have suggestions as to how to approach it?

[Kevin_Sevcik]

1. and 2. of your system are incorrect. The left hand side is correct, but the right hand side is wrong. What is the derivative of (x-1)^2 with respect to x? Ditto (y-1)^2 w.r.t. y. You might try fully expanding the expression and then deriving it if you’re still getting 2x and 2y.

Hint: The chain rule always applies in derivatives, everywhere, all the time.

[Astronouth7303]

1. and 2. of your system are incorrect. The left hand side is correct, but the right hand side is wrong. What is the derivative of (x-1)^2 with respect to x? Ditto (y-1)^2 w.r.t. y. You might try fully expanding the expression and then deriving it if you’re still getting 2x and 2y.

Hint: The chain rule always applies in derivatives, everywhere, all the time.

Darned me and those shortcuts. I didn’t think the constant offset would change it much.

Ok, it doesn’t. Just 2(x - 1) instead. It’s always the little things.