Updated with video links and a curriculum clarification for the portfolio separation.

The "20 or 21" videos that ended up being the beginning of "21".

the C¹ functions characterization, https://youtu.be/v6MeRFSL0YM
the characterization for only continuity assumptions, in terms of generalized gradients (supergradients for concavity, subgradients for convexity), and sketchy: the supporting-line characterization of quasiconcavity/quasiconvexity (the "price vector"), https://youtu.be/t6JxIaP-UJM

The "fact sheet" lectures

The PDF first: https://folk.universitetetioslo.no/ncf/4140/lectures2021/concavity+quasiconcavity_FACTSHEET.pdf
"Page 0" is just some picture, so the second actual page in the PDF is fact sheet "1". That makes "1" through "4".

Fact sheet 1: The basics. https://youtu.be/ATzJbd4Wkfw
Fact sheet 2: General properties, https://youtu.be/rOZCjSx9IAw
There is an example and proof-sketches in a separate clip: https://youtu.be/y9udRAkQrHc
Fact sheet 3: Special cases and properties for (unconstrained) optimization. https://youtu.be/JYif8YR03XY
For one of these, the concavity of homogeneous positive degree-1 homogeneous functions, see also this Canvas document. And, the application to Cobb-Douglas here.
Moved to lecture 22: Fact sheet 4. https://youtu.be/Lpbve9ZK_vY

More lecture 22: concave programs examples

Three examples - a fourth (exam 2015 no. 2) will be given as seminar problem. And then the portfolio separation video at the very end, the "textbook version" of the problem will be a concave program.

One example here: https://youtu.be/dCQHvM4GGlU

Two more examples here: https://youtu.be/uv_m8gcxhSw

The final optimization lecture:

Precise Lagrange (& Kuhn-Tucker) conditions. https://youtu.be/mb9VjyGdBWw
- First part until 15:45 (with four more minutes of talk): the main theory.
- From 19:51: nine minutes on what scenarios the constraint qualification would fail.
- From 28:45: Deducing the (precise and the "Math 2") Lagrange conditions.
A portfolio optimization problem (a classic from theoretical finance!): https://youtu.be/T3NJI04ctoo
The risk-averse agent has a concave program, while the more general agent will get a problem that is not up-front concave, and where one of the (not-so-awful) cases of CQ failing will materialize in a special case.
Curriculum clarification:
- The finance part of this is not Math 3 syllabus, nor is the probability part
- Doing the math is Math 3 syllabus. That includes solving in matrix form, recognizing the concave program and its consequences for sufficient conditions, and also manipulating (again in matrix form) the max [linear] subject to [quadratic] that is not ex ante concave.

Some considerations beyond Math 3?

Sadly (?) I cannot overload curriculum with everything an economist should know and that is even curriculum in compulsory courses: Micro 3 goes beyond Mathematics 3 on a few on optimization topics. For example the duality between utility/output maximization subject to budget, and cost minimization subject to utility/output.

It is not mathematically obvious that max output s.t. budget and min expenditure s.t. output should give the same allocation and cost. But the conditions for that are related to convex sets.
The same goes for a Nash equilibrium for a zero-sum game: zero-sum means that I want to minimize your payoff (not out of ill will, but because that is my gain), and we are looking at max_x min_y criteria. Minimax or maximin. For a Nash equilibrium, it shouldn't matter who "has the last word", because the first player anticipates the second. But that depends on the problem actually admitting nice properties; lo and behold, the conditions should look familiar, at least if we work on continuous functions in Rⁿ: https://en.wikipedia.org/wiki/Sion%27s_minimax_theorem
- There are actually minimax properties lurking in the Lagrangian $LaTeX: f(\mathbf x)-\lambda (g(\mathbf x)-b)$ : in fact, part of the K-T conditions are that λ minimizes the Lagrangian. (The λ coefficient $LaTeX: b-g(\mathbf x)$ is nonnegative, and if it is positive, then λ = 0 maximizes over all nonnegative λ.)
And as hinted on in a video: "pure" strategies are "either-or". Randomizing between A and B gives a convex opportunity set, the interval [0,1] of probabilities. You have utilized this for a long time already.