Let's have the periodic function
where $t_0$, $t_1$ and $t_2$ are equally spaced. All I need to know is the sign of the sum
Obviously, however, the above function is not differentiable and therefore $f′(t_i)$ cannot be determined. Can I substitute the above sum by the product of the average function value for the period and its second-order divided difference and thus make a conclusion about the sign of that sum -- say, in the case at hand claim that it's negative?