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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 $$ \frac{f(x)-f(x_0)}{x-x_0}\begin{cases} >0 & \text{if }xx_0. \end{cases} $$ This difference quotient is undefined when $x=x_0$, but mere algebra tells us that the numerator factors and we get $$ \frac{(x-x_0)g(x)}{x-x_0} = g(x) $$ where $g(x)$ is a polynomial whose coefficients depend on $x_0$. Then of course one seeks its zeros since it should change signs at $x_0$.

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!)

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    +1 for a question tagged [tag:calculus] that tries to avoid calculus :-)2012-09-21
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    Michael Livshits has done a pretty thorough and systematic presentation of this approach: http://www.mathfoolery.com/calculus.html Marsden has a book called Calculus Unlimited, which I don't feel was very successful: http://www.cds.caltech.edu/~marsden/books/Calculus_Unlimited.html One of the awkward issues in carrying this very far is that you end up having to do a lot of special-casing for, e.g., points of inflection. This gets tedious in the Marsden book.2012-09-21
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    In my pre-calculus days I mastered many beautiful inequalities and used them to solve this sort of maximisation problems, which could be challenging but fun. After the introduction of calculus I just take the derivative and compute values of functions at critical points, a technician's job. What a de-mathematician process!2012-09-22
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    Currently I am teaching from some notes of Korner which would argue that since similar arguments can fail when working with functions from the rationals to the rationals, some analysis (Extreme Value Theoren I guess) is needed. However, if one takes the POV that local maxima always exist for such "reasonable" functions, and that the challenge is to find them, then it seems plausible one can do it for polynomials without the standard machine of calculating the derivative and setting it to zero, etc2012-09-24

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