I know you can just use derivatives, but is there a way to solve for them without doing that? Is graphing it the only other way? Thanks in advance.
Sorry about the spelling errors. Can a mod please change those if possible?
I’ve never seen anything like that. Why would you want to?
Well.
This is stupid, but it would work I think.
Take the integral of the function. This allows you calculate the area under the curve from point A to point B.
Use a computer to calculate the area from X=A to X=B. Now calculate from X=A to X=B plus some small number. Do that over and over adding some small number everytime.
That should allow you to detect inflection points without technically graphing the function or taking a dir.
Was just heading out the door, so havent given this much thought.
Define “using derivatives”. For instance, do you consider the Secant method to be using derivatives?
I don’t understand the intuition behind this at all.
Try doing this. Say you have a function from a bounded subset of R to R. Split the domain of the function into n intervals, and only consider the value of the function at the endpoints of your intervals. For each endpoint, check the value of the function there and at both neighbours. If both neighbours have a higher (resp. lower) value, you have a minimum (resp. maximum). Otherwise it was a regular point. The finer the mesh you’re using, the closer you are to the actual critical point set of your function. This is basically how it’s done in numerical analysis, when you can’t always symbolically differentiate your function.
Similar but more complicated things could be done to find critical points of functions from R[sup]n[/sup] to R[sup]m[/sup].
Of course, for certain specific types of functions (like quadratic functions and sine waves), it’s possible to give a formula for where the critical points or inflection points are.
Well, to check for changes in concavity (which is what inflection points indicate), you could use the definitions of a convex and concave function. A concave function, by definition, lies above a line connecting two points on the graph. So pick some interval [a,b] and a mesh size, say, [b-a]/10and see if that condition is true for a, a+[b-a]/10, …, b. If the condition changes somewhere in that interval, there’s an inflection point somewhere. Rinse and repeat, maybe with smaller mesh.
The reason to use derivatives to find extrema and inflection points is that other methods will only give approximations (but very good ones). Derivatives, though, are based on the idea of limits which, due to certain properties of the real line, can give you answers to an arbitrary degree of precision, and are in that sense “exact.”
[btw: there’s probably some problems with what I said, but I’m not a mathematician.]
What I see as a problem with this is that for a function f(x), in any interval [a,b] where a,b are reals, there could be an arbitrary number of local extrema (minima and maxima) and inflection points, so you are looking at recursing infinitely over smaller and smaller intervals looking for tiny tiny little twitches, and voila, you end up needing to use limits to avoid having to chug f(x) for aleph(1) reals eternally, and you are back at the deriviative.
E.g. you could say that you are going to measure intervals that are 0.0001 units wide, assuming that’s small enough, but then some wise guy gives you a function that for the interval [0.0000000000000004 0.0000000000000005], has 50 local extrema.
The issue with this approach is that the number of grid points you need to check (and the number of neighbors of each of them) grows exponentially with the dimension of your domain. It works well in one or two dimensions, but after that, you’re not going to want to run it over a particularly fine grid.
This isn’t my area of expertise, but once you’re dealing with functions whose codomain is not totally ordered, you probably have to be very careful about specifying exactly what you’re looking for. Wikipedia says that the notion of a critical point doesn’t generalize in the completely obvious way, and I have no idea how to search for points like that. (That doesn’t mean that there isn’t an efficient algorithm, mind you, but it’s not going to be obvious to a non-specialist.)
Just dropped in to note that I like the word “finction.” I’m going to have to come up with a definition, though.
It’s a made-up function that you specifically tweaked to demonstrate some particular property that you wanted to illustrate for the students, although you told the class that you were just selecting some function at random to use in your example. 
Pedantic note: aleph_1 isn’t the cardinality of the reals, except under the Continuum Hypothesis. beth_1 is the cardinality of the reals.
Nobody’s pointed out you can just graph the thing and pick them out? Sure, it doesn’t have the precision of calculus, but in lots of real world engineering examples it’s plenty good enough
A critical point of a function is a point where its derivative is neither positive or negative; an inflection point of a function is a point where its second derivative changes sign. No one’s forcing you to take derivatives to figure these out (any correct reasoning to find them is, you know, correct), but generally people have lots of practice taking derivatives and lots of practice finding points where functions are neither positive or negative, or points where they change sign, so taking advantage of this is, in addition to being quite directly motivated, often an easy way to go. As ultrafilter asked, what situation do you have in mind for which you are opposed to using derivatives?
The OP did, when he asked if graphing was the only alternative.
Just got back home.
If your function has inflection points, the area under the curve will have inflection points of a sort as you plot total area vs X.
Its kind of a “why bother?” and stupid way to do it, but it can probably dectect at least some kinds of inflection points without technically graphing the function itself or taking the dir to directly compute inflection points.
I don’t understand billfish678’s method either. You’re basically numerically computing the integral of f; how is it helpful to move from a function whose second derivative’s changes of sign are the points of interest to a function whose third derivative’s changes of sign are the points of interest? It’s all the bother of moving into the derivatives, except only numerically and in the other direction!
You’re right, I should have said R[sup]n[/sup] to R. I believe I’ve seen an extension of critical points to functions from R[sup]n[/sup] to R[sup]m[/sup], but it’s been a long time so I don’t remember.
No, I said TAKE the integral, then use THAT to calculate area vs changing X and blah blah blah.
Now, if its an equation you can’t find the exact integral of, you’ve got problems.
And yeah, I said it was stupid, but it is (assuming my thinking isnt all buggered up here) a way to get at least some inflection points without graphing the function itself or explicitly taking the dir.