I am preparing some sheets of exercises that I'll assign to my undergraduate students in biology (sophomore class, or first academic year in italian universities). This is the problem:

Exercise. Let $f \colon [a,b] \to \mathbb{R}$ be a continuous and convex function. If $f(a)f(b)<0$, prove that $f$ has exactly one zero.

The solution is essentially clear from the graph of $f$, but I wish they could supply a more rigorous proof. According to your experience, is this problem too hard for this kind of students? Should I be satisfied with a "graphical" answer? Apart from the geometric and analytic definition of convexity, what properties of convex functions should they kknow, to solve rigorously this problem?

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I think they should be able to translate the picture into a statement. In particular, it is almost trivial to show that if $f$ has two zeros, then $f$ cannot be convex (under the conditions stated). The picture makes this obvious both graphically and analytically (in my opinion). They need only know the usual definition of a convex function (ie, $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda) f(y)$). –  copper.hat Aug 18 '12 at 8:58
Well. From recent experience with non-math majors I can predict that many students will have problems to see that they have to show that if there are two zeroes, then the function is not convex. This kind of thing seems trivial to people experienced in proving things but apparently it is not. I look at the problem and think "it's clear that there is at least one zero. Why can't there be more than one". These little steps of clarifying the situation seem to be very difficult if you are not used to doing this. –  Stefan Geschke Aug 18 '12 at 9:15
It is very hard to accept one particular answer, since they reflect a personal viewpoint. I strongly appreciate your contributions, though they tend to underrate the level of students. My course is more than basic calculus, since I teach theorems and proofs. My students learn the theorem about zeroes of continuous functions, but I am sketchy on the theory of convex functions. –  Siminore Aug 18 '12 at 11:07
Perhaps give them a hint as in what happens if you take $f(a)<0, f(b) >0$ and two points in between with $f(x) = f(y) = 0$? Have them draw a picture. –  copper.hat Aug 18 '12 at 17:46

Yes, this problem is almost certainly too hard. In my experience you can be happy if the students get any graphical intuition at all (for problems where functions are involved that are not given by an explicit formula). You won't get a rigorous proof. When I taught calculus (mostly engineering students) the only proof that some of the student were ready to give were proofs that followed exactly the same pattern as other proofs done in class before. I would think that already decoding $f(a)f(b)<0$ as "one is $>0$, the other is $<0$" is a challenge. This is usually different with math majors, but students with other majors often have difficulties with these things, in my experience.

What would be needed to give a rigorous proof? Certainly the intermediate value theorem (to get at least one $0$). Everything else could be done using just the definition of convexity.

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The only property of convex functions needed is that the graph of the function cannot lie above the chord between any two points on the graph, which is basically the definition af a convex function. Nevertheless, there are many potential stumbling blocks here, each of which might be expected to take its toll among the students.

The condition $f(a)f(b)<0$ has to be interpreted as saying that the function values at the two endpoints have different signs, and the problem has to be divided into two cases. To show that the function cannot have more than one zero in the interval $[a,b]$, one have to assume the existence of two zeros, put names on them, such as $x_1$ and $x_2$, and use these points in the argument. My experience is that student are often insecure about such matters. "Are we allowed to do that?" is a typical question. This is particularly true of students who do not major in mathematics, and for whom mathematics is often just a toolbox of techniques to solve certain kind of problems that one have to deal with.

Another potential problem is that for many students, convexity means no more or less than $f''(x)\geq 0$ for all $x$. What to do when there isn't any expression to differentiate?

If you are afraid that your problem will be too difficult for your students, it is possible to make it easier by explicitly saying that $f(a)<0$ and $f(b)>0$, and/or giving a hint, such as "For showing that the function cannot have more than one zero, use the definition of a convex function (on three suitably chosen points)".

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