We're trying to minimize the function
, subject to the restriction
, which should immediately suggest the Lagrange multipliers method.
Of course, it's easier to work with
. We can minimize this function, then take a square root. Then, letting
, we set
, so we have
We can quickly work out that the first two equations are satisfied when either
can never satisfy
, so we must have
, and thus
This is the "this is a timed test and we're in a hurry" version. I'm sure someone else could give a more lucid explanation.