Do we really need coercivity to have minimizers/maximizers? Looks like extreme value theorem should be sufficient.
By "two sufficient conditions" do you mean they are both sufficient, so either one being true will guarantee a minimum, or are they both required? Because I'm pretty sure continuity over a compact domain by itself is enough, but coercivity by itself is not (considering x^2 with the zero removed).
How do we define coercivity mathematically? An objective function f such that for every c there exists some x such that |x'| > x implies f(x') > c?
Why is coercivity relevant in this case? I don't think it affects existence of minimizers.
Why does the objective function going to +- infinity give us the existence of a minimizer? Couldn't we have that the function e.g. becomes 1 after a certain x coordinate (so the limit still exists but isn't infinite) – wouldn't that allow us to still find a minimizer? And if 1 is the minimum value, we could prioritize the first x-coordinate that gives us this value.
Is it realistic to necessitate continuity for graphics problems? I feel that many problems such as measuring velocity (with sudden changes) would be difficult to do so.
Do we need to formulate optimization problems under these conditions? Or is it also ok to form them under other conditions that make them feasible or even under conditions that do not ensure feasibility?