How to tell if second derivative is positive or negative if there are no numbers in it?

It's similar to how you might solve an inequality involving a polynomial. For example,



First consider the case when the left side actually is zero. This happens when and . Pick numbers that fall to either side of these values and plug them into the inequality.

For instance, take (to the left of -1), (between -1 and 2), and (to the right of 2). You have





So the polynomial is positive for and .

You can use a similar analysis to determine the sign of the second derivative over certain intervals.