I need to use Lagrange multipliers to find the maximum and minimum values of function f(x,y) = x^2 + y^2 + z^2 subject to the constrain function xyz = 1.
This is what I have so far:
Maximize and Minimize f(x,y,z) = x^2 + y^2 +z^2
Subject to: g(x,y,z) = xyz-1
F(x,y,z,λ) = f(x,y,z) + λg(xyz-1)
Fx = 2x + λyz = 0
Fy = 2y + λxz = 0
Fz = 2z + λxy = 0
Fλ = xyz – 1 = 0
Fx = 2x = -λyz x = -λyz/2
Fy = 2y = -λxz y = -λxz/2
Fz = 2z = -λxy z = -λxy/2
Then this is where I am stumped. If I multiply all of these together I have
-1/8 λ^3 x^2 y^2 z^2 = 1 I am fairly sure I am headed in a wrong directions
Any help would be appreciated.
Thank you
Emily
Best Answer
Notice that your constraint is unbounded, and that $x=n$, $y=1/n$, $z=1$ is a solution for all $n$. It follows that the maximum is infinity, by taking $n\rightarrow\infty$.
For the minimum, as a hint, deduce that $2x^2+\lambda=0$, $2y^2+\lambda=0$ and $2z^2+\lambda=0$. Can you finish it from here?