Question

Obtain the shortest distance from the origin to the sphere xyz^2=2

Answer 1

Define f(x,y,z)=x2+y2+z2−−−−−−−−−−√f(x,y,z)=x2+y2+z2, which is what we want to minimize, with the constraint g(x,y,z)=xyz2=2g(x,y,z)=xyz2=2.

We get three equations (I'm using ff as shorthand to avoid the messy root):

xf=λyz2xf=λyz2 or x2f=2λx2f=2λ

yf=λxz2yf=λxz2 or y2f=2λy2f=2λ

zf=2λxyzzf=2λxyz or z2f=4λz2f=4λ

We get y=±xy=±x, z=±2–√⋅xz=±2⋅x and with the constraint only y=xy=x will work.

Finally we get x=±1x=±1 and this gives us 44 points ±(1,1,2–√)±(1,1,2) and ±(1,1,−2–√)±(1,1,−2).

I haven't really used λλ though... so it's like I didn't really do it properly.