Ohio Assessments for Educators (OAE) Mathematics Practice Exam

If the second derivative is greater than zero, what type of critical point is it?

1. Maximum

2. Minimum

3. Saddle Point

4. None

The correct answer is: Minimum

When the second derivative of a function is greater than zero at a critical point, it indicates that the function is concave up at that point. This means that the graph of the function is curving upwards, and that there is a local minimum at this critical point. To understand this concept, consider the implications of the first and second derivatives. The first derivative tells us about the slope of the function at a particular point. When the first derivative equals zero, it indicates a potential maximum, minimum, or saddle point—known as a critical point. The second derivative, however, provides further insight into the nature of that critical point. A positive second derivative implies that as you move slightly left or right from the critical point, the function values increase, confirming that this point is indeed a local minimum. Overall, a critical point with a second derivative greater than zero signifies that the function is experiencing a minimum value, making it a crucial point of interest for analyzing the function's behavior and graphing.