# How would I find the minimum of the sum of $x, y, z$, given the product that $xyz=27$ using Lagrange Multiplier?

Find three positive integers x, y, z that satisfy the given conditions. The product is 27, and the sum is a minimum.

I'm lost on how I would solve my system of equations that I have set up.

$$1=\lambda \cdot yz$$

$$1=\lambda \cdot xz$$

$$1=\lambda \cdot xy$$

$$xyz=27$$

Are my system of equations correct?

Yes, the system is$$\left\{\begin{array}{l}1=\lambda yz\\1=\lambda xz\\1=\lambda xy\\xyz=27.\end{array}\right.$$It follows from any of the first $$3$$ equations that none of the numbers $$x$$, $$y$$, and $$z$$ can be $$0$$. So$$1=\frac11=\frac{\lambda yz}{\lambda xz}=\frac yx;$$that is, $$y=x$$. By the same argument, $$z=x=y$$. Can you take it from here?