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?