More coffee

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Two young mathematicians witness the perils of drinking too much coffee.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley!
Riley: Yes Devyn?
Devyn: Do you like coffee? I like coffee! Sometimes I feel really ‘‘bad,’’ sluggish and tired. Then I drink coffee and I feel good! Sometimes I drink a lot of coffee!
Riley: Um?
Devyn: But here’s the problem, see: If I drink too much, I become over excited and can’t stop talking. I just drink coffee, then talk. Then drink more coffee. Then I start to feel sick. Ugh. I have a love-hate relationship with coffee.
Riley: If only there were a calculus solution to this problem!

Remember, calculus is about studying functions. If we can ‘‘see’’ a function in the work above, maybe we can figure out how to solve it.

If we were to try to solve Devyn’s coffee problem, what would be the best function to know?

How many donuts Devyn eats. How ‘‘good’’ Devyn feels after $x$ cups of coffee. How many cups of coffee Devyn drinks when Devyn feels $x$ ‘‘good.’’ Impossible to say.

If we let $f(x)$ be ‘‘How ‘good’ Devyn feels after $x$ cups of coffee.’’ And we think about what Devyn says above, is there an amount Devyn can drink and feel maximally ‘‘good?’’

yes no

‘‘Yes’’ is probably the best answer, though we are assuming that Devyn feels ‘‘bad’’ without enough coffee, ‘‘good’’ with some, and ‘‘bad’’ again with the function continuously growing and then decreasing.