A mysterious formula

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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 $f(x) = \frac {10}{x^2-2.8x+3}?$
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 $[0,10]$ , 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 $[0,10]$ to be? The maximum is at $x=\answer [tolerance=.2]{1.4}$

If you wanted to argue that this is the (global) maximum value on $[0,10]$ without plotting, what arguments could you use?