You could try a LaGrange multiplier.
We want to find the extrema of subject to the constraint
Therefore, we have
Now, you can set  and  equal and solve for y. Set  and  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.