First, and we have already touched upon: The C² functions characterization. Also I found a webpage which is quite nice on several topics including this: https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/toc , see section 3.4 for quasiconcavity.

a: The basics recapped, and "when does graphical max-subject-to-budget solutions work"

18 min, some of you may speed up this one, but I will probably refer to it later: Recapitulates the basics and also the graphical solutions that you have been doing since the dawn of time: what does it take for that solution method to be valid? Roughly speaking, quasiconcavity is what you want for the max problem. These sketches do not really depend on the smoothness of the functions, and could be recycled for the C¹ characterizations.

https://youtu.be/p43LDk10vMk

b: The actual theory part for C² functions: bordered Hessians.

13 minutes, youtube video 20b: https://youtu.be/nQ-HD_TvrLE

c: Example - the quasiconcavity of the bivariate Cobb-Douglas.

Example time! To no-one's surprise, we start with the (strict!) quasiconcavity of the bivariate Cobb-Douglas where we only assume positive exponents (so the function need not be concave!).

Note, there are easier ways: most notably, the Cobb-Douglas is exp([a concave function])!

Youtube video 20c, 17 min: https://youtu.be/UEmww-P2aaY

d: Example - the n-variate Cobb-Douglas.

The strict quasiconcavity of the n-variate Cobb-Douglas, in terms of bordered Hessians (despite that not being a very efficient method. The next couple of lectures will have much less "Math 2" calculations ...)

19 minutes, YouTube video 20d: https://youtu.be/MOu9rexwEYw

e: Relation to level (hyper-) surfaces & more.

You probably have seen quasiconcavity explained by "decreasing convex level curves". (That requires the preference direction to be north-east!) Here follows how to do calculations on the upper level sets and the level curves determining one variable as convex function of the others. Note again to the end: there are simpler ways than calculating determinants for this example too.

9 min: Youtube video 20e, https://youtu.be/pfnOtRBIAx8