Some people may carelessly say that you need calculus to find such a thing as a local maximum of $f(x) = x^3 - 20x^2 + 96x$. Certainly calculus is sufficient, but whether it's necessary is another question.
There's a global maximum if you restrict the domain to $[0,8]$, and $f$ is $0$ at the endpoints and positive between them. Say the maximum is at $x_0$. One would have $ \frac{f(x)-f(x_0)}{x-x_0}\begin{cases} >0 & \text{if }x
Have we tacitly used the intermediate value theorem, or the extreme value theorem? To what extent can those be avoided? Must one say that if there is a maximum point, then it is at a zero of $g(x)$? And can we say that without the intermediate value theorem? (At least in the case of this function, I think we stop short of needing the so-called fundamental theorem of algebra to tell us some zeros of $g$ exist!)