Multivariable Optimization - Closest point on surface

[theWapiti]

Homework Statement

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

Homework Equations

is the distance along PQ.

EDIT:

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.

 

[Mentallic]

Homework Statement

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

Homework Equations

is the distance along PQ.

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.

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.

 

[theWapiti]

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.

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?!

 

[theWapiti]

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