how to show that a function is convex over a range?

Hi,

the following is not really maxima related, but had no better idea where to
ask, and my economics background is not enough for this type of math :-\

I have a rather complex polinomial of which I am interested in the concave
roots on the interval [0,1].

Maxima was not able to solve it for general roots, and when I've plotted its
derivative with some parameter values over the relevant range it always
showed me an increasing, function, thus having corner solutions. Are there
some mathematical methods, ideas, best practices to verify that a function
is indeed convex over a given range for a fairly large set of possible
parameter combinations?

Any references would be appreciated.

thanks for your help,
Viktor