Lecture 02: Gm-C filters from a signal-flow point of view
This exercise was formulated in a manner that made it possible to understand it as "calculate the optimum". This was an unfortunate mistake of Hanspeter's; what he meant was "use a small set of formulas and then play with Matlab by hand to get to know the problem.
Georg really went through all of it analytically, a very commendable effort to be found below as the reference solution. Using the concept of the noise bandwidth, the solution can be shortened to the alternative reference solution written by Hanspeter.
However, the point was simply to let you find out that if you get the task to optimize noise without having a constraint on the gm, you can make that noise as low as you please.