Here's one for ya.

Show that if $\displaystyle f$ is differentiable on an open interval $\displaystyle I$, and if the graph of $\displaystyle f$ is concave upward on $\displaystyle I$, then the graph of $\displaystyle f$ lies above all of its tangent lines on $\displaystyle I$.

This is one of the problems given in my text. Although it is not required that I answer it, I feel that this problem could deepen my understanding of the second derivative test. Intuitively, I can visualize this, but I'm having trouble putting it into the language of math. Anybody got any Ideas?