
Two young mathematicians discuss optimization from an abstract point of view.

Check out this dialogue between two calculus students:
Devyn
Riley, what do you think is the maximum value of
Riley
Where did that function come from?
Devyn
It’s just some, um, random function.
Riley
Wait, does this have to do with coffee?
Devyn
Um, uh, no?
Riley
Well what interval are we on?
Devyn
Let’s say $$, I mean there’s no way I could possibly drink ten cups of coff…
Riley
I knew this was about coffee.

Here Devyn has made a function, that is supposed to record Devyn’s ‘‘well-being’’ with respect to the number of cups of coffee consumed in one day.

Graph Devyn’s function. Where do you estimate the maximum on the interval $$ to be? The maximum is at $$
If you wanted to argue that this is the (global) maximum value on $$ without plotting, what arguments could you use?