# Multivariable Optimization - Distance Formula, Use of Square Root

I'm working through a problem in which I am trying to find the point $P(x,y,z)$ closest to the surface $f(x,y)$. I am not concerned with the actual distance, I just want to find the closest point $P$.

To do this I am minimizing the distance between the surface and the point using the standard distance formula $\sqrt{x^2 + y^2 + f^2}$.

I think believe however that the square root is not needed, for I am only concerned with the closest point, not the actual distance.

Question: Do I need to include the square root?

I am very confident that I don't need it, but I just wanted to make sure.

Thanks

## 2 Answers

You are correct in that you don't need the square root:

If $\sqrt{h(x,y,f)}$ is at a minimum, then $h(x,y,f)$ is at a minimum, for $h\geq 0$.

No. The argument you have is nonnegative, and the square root function is increasing for positive $x$, so $\sqrt{g(x,y)}$ has a minimum if and only if $g(x,y)$ has a minimum.