You could try a LaGrange multiplier.

We want to find the extrema of subject to the constraint

Therefore, we have

Now, you can set [1] and [2] equal and solve for y. Set [1] and [3] equal and solve for z. Then sub them into the constraint to find your x value. Then you can sub that back into the other equations to find y and z.

Plug those into f(x,y,z) to find the point that is closest the origin.

There may be several to check.