# Multivariable Optimization - Closest point on surface

1. Jul 7, 2014

### theWapiti

1. The problem statement, all variables and given/known data

Find the first-octant point P(x,y,z) on the surface closest to the given fixed point Q (0,0,0).
The surface x2y2z=4

2. Relevant equations

is the distance along PQ.

EDIT:

3. The attempt at a solution

I get stuck here every time. I feel like I'm just selling myself short here, but I don't know how to resolve the situation for when the critical point has a variable in it.

Last edited: Jul 7, 2014
2. Jul 7, 2014

### Mentallic

Your derivative is incorrect. Try again, more slowly this time.

When you've done that, then take the partial with respect to y (this should be easy because the problem is symmetric, so you just need to swap the x and y variables in your partial with respect to x) and set that equal to 0 as well. You then have two equations in two unknowns.

3. Jul 7, 2014

### theWapiti

I feel like I'm probably just making myself problems here, but I can't for the life of me get past this still.

I was minimizing for the distance squared, which gave the two partials. But how can I solve if I have y in terms of x and x in terms of y?!

4. Jul 7, 2014

### theWapiti

Well I feel like a dummy. Got it now, obviously. Sometimes the problem is made so much more difficult in your own mind!