Lagrange Multipliers

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?

  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.