I'm supposed to find the maximum and minimum value of $x^2+y^2+z^2$ subjected to condition $1/x +1/y+1/z=0$.
What trips me is when ending with a λ = $0$
if λ = $0$, then all $x, y, z$ values become $0$.
My book gives an answer as $Min = 27$
Is this correct because I can't arrive at that particular value.
If $Min = 27$ then $x = 3, y=3 ,z=3$ should be the critical point.
The Lagrange condition says
$$2x=-\lambda/x^2 \\ 2y=-\lambda/y^2 \\ 2z=-\lambda/z^2$$
which means $x^3,y^3$ and $z^3$ are all the same number, which in turn means that the same is true of $x,y,z$, since $x \mapsto x^3$ is a one-to-one function. But there is no such point on the constraint surface (such a point would have $\frac{3}{c}=0$ which has no solution), so the Lagrange condition is never satisfied.
In terms of a global max/min, you can see that a global min can't exist by looking at points on the constraint surface close to $(0,0,0)$, e.g. $x=h,y=-2h,z=-2h$ as $h \to 0$. Similarly you can see that a global max doesn't exist by looking at points on the constraint surface where one of the variables is very large.
My guess is there is some error in the problem statement, especially since the proposed answer isn't on the constraint surface. One possibility that would line up with the proposed answer would be that the constraint was meant to be $1/x+1/y+1/z=1$ rather than $0$.