Lagrange multipliers...what is my constraint?

Question

How would I use Lagrange multipliers to determine which point on the surface $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ with $x,y,z>0$ is closest to the origin?

I'm not sure what the constraint would be or how I would approach this. Any help is appreciated.

Thanks

Answer 1

Hint:

You want to minimize $$ f(x, y, z) = x^2 + y^2 + z^2 $$ while under the constraint $$ g(x,y,z) = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 $$ So you should set $$ \nabla f = \lambda \nabla g $$ i.e. $$ (2x, 2y, 2z) = \lambda \left(-\frac{1}{x^2}, -\frac{1}{y^2}, -\frac{1}{z^2} \right). $$