Question

Desperately need help with Langrangian optimization with inequality constraint!!! Please, please, please!!


max (x,y,z) 12(x-y-z)-3x^2-4y^2-6z^2 s.t. x^2+y^2+z^2=<400

L 12(x-y-z)-3x^2-4y^2-6z^2-(lambda)x^2+y^2+z^2-400

dL/dx = 12-6x-(lambda)2x = 0 -> 12-6x=2x(lambda)

dL/dy = 12-8y-(lambda)2y = 0 -> 12-8y=2y(lambda)

dL/dz = 12-12z-(lambda)2z = 0 -> 12-12z=2z(lambda)

dL/dlambda = x^2+y^2+z^2-400 = 0

Answer 1

use standard methods to find local max/mins then reject any that don't satisfy

\[x^2+y^2+z^2\le400\]

then use Lagrange multipliers to check the boundary

\[x^2+y^2+z^2=400\]

Answer 2

I've gotten to the point of setting up the Lagrangian, but I can't figure out the solutions.

Answer 3

why don't you show what you have so far

Answer 4

Good point. I'll edit the question. Thank you!

Answer 5

your dL/dy and dL/dz are a little off

Answer 6

\[\frac{\partial }{\partial y}(12(x-y-z))=-12\]

Answer 7

You take the partial derivative. X and Z drop off.

Answer 8

\[\frac{\partial }{\partial y}(12(x-y-z))=\frac{\partial }{\partial y}(12x-12y-12z)\]

\[=\frac{\partial }{\partial y}(12x)-\frac{\partial }{\partial y}(12y)-\frac{\partial }{\partial y}(12z)=0-12-0=-12\]