Minimise $f(x,y,z)=x^2+y^2+z^2$ subject to the condition $1/x+1/y+1/z=0$

I've been trying to solve this question using Lagrange's Multiplier method, using which I get the following equations:

$2x$-$\lambda\over x^2$$=0$

$2y$-$\lambda\over y^2$$=0$

$2z$-$\lambda\over z^2$$=0$

Which reduces to:

$\lambda=x^3=y^3=z^3$

Which implies $x=y=z$

Plugging this back into the condition gives:

$1\over 3x$$=0$

I cannot seem to be able to solve this system of equations.

Solution 1:

The infimum of this function is $0$ and it is not attained. Take $x=y=\frac 1 n$ and $z=-\frac 1 {2n}$ to see that the infimum is $0$. You got into trouble because there is no minimizing point $(x,y,z)$.