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.


  • 3
    $\begingroup$ The constraint is the equation you have written down (later you can worry about the $x,y,z\gt 0$ part). The function you are trying to minimize is $\sqrt{x^2+y^2+z^2}$, or, better, $x^2+y^2+z^2$. $\endgroup$ – André Nicolas May 8 '14 at 18:18


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. $$ (2x, 2y, 2z) = \lambda \left(-\frac{1}{x^2}, -\frac{1}{y^2}, -\frac{1}{z^2} \right). $$


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