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)".