Undergraduate approach to a problem about convex functions 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?
 A: 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)". 
A: 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.
