We have :

      x^2+xy+y^2=3 and
      y^2+yz+z^2=16
      A=xy+yz+zx

Find the maximum value of A. Find x, y and z when A=max value.

(Remember the category)

The Lagrange multiplier theorem states that a function is maximized subject to several constraints when grad[f(p)] = L1 grad[g1(p)] + L2 grad[g2(p)]..., where p is a vector quantity.

In this case,
grad[g1(p)] =
grad[g2(p)] =
grad[f(p)] =

Substituting into the Lagrange multiplier equation, we get a bunch of equations:

y + z = L1(2x + y)
x + z = L1(x + 2y) + L2(2y + z)
x + y = L2(y + 2z)

Plus two constraint equations:
x^2 + xy + y^2 = 3
y^2 + yz + z^2 = 16

That gives us a yummy five equations with five unknowns. Solving for L1 and L2 in the first and third equations, we get:

L1 = (y + z) / (2x + y)
L2 = (x + y) / (y + 2z)

Substituting into the second equation, we get:

x + z = (y + z)(x + 2y) / (2x + y) + (x + y)(2y + z) / (y + 2z)

Simplifying a bit,
(x + z)(2x + y)(y + 2z) = (y + z)(x + 2y)(y + 2z) + (x + y)(2y + z)(2x + y)

(2x^2 + xy + 2xz + yz)(y + 2z) = (xy + 2y^2 + xz + 2yz)(y + 2z) + (2xy + xz + 2y^2 + yz)(2x + y)

2x^2y + xy^2 + 2xyz + y^2z + 4x^2z + 2xyz + 4xz^2 + 2yz^2 = xy^2 + 2y^3 + xyz + 2y^2z + 2xyz + 4y^2z + 2xz^2 + 4yz^2 + 4x^2y + 2x^2z + 4xy^2 + 2xyz + 2xy^2 + xyz + 2y^3 + y^2z

-2x^2y + 2x^2z - 6xy^2 - 2xyz + 2xz^2 - 4y^3 - 4y^z - 2yz^2 = 0

Dang, I was hoping SOMETHING would cancel out of that. I gotta get ready for work. Anybody want to check my work and/or take it from there?

Posted by friedlinguini on 2002-06-19 04:18:57