Question: Determining convexity of functions

I'm an upper-level undergraduate mathematics student, and I'm working on a project to create an algorithm to determine if any function is convex.  My Professor suggested that I post in this forum, to appeal for suggestions.

I hope to apply the algorithm to single or multivariable functions, which have polynomial, power, logarithmic, exponential, trigonometric, or any combination thereof.  The algorithm should work on any predefined domain.

My general strategy thus far has been to consider the Hessian determinant or the characteristic roots of the Hessian. However, these approaches are much simpler when evaluating a function at a critical point.  To use them to determine convexity I essentially need to know if a function (e.g., the Hessian determinant) is always positive.

I hope to apply the following facts about positive functions:
if f(x) and g(x) are positive then

f(x)+g(x) is positive,
f(x)*g(x) is positive.

Any suggestions?  

Big thanks,

Paul Burkander