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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, $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda) f(y)$).2012-08-18
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    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.2012-08-18
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    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.2012-08-18
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    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.2012-08-18

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