Ohio Assessments for Educators (OAE) Mathematics Practice Exam

The second derivative test is primarily used to determine what?

1. The nature of critical points

2. The value of the function

3. The slope of the function

4. The intercepts of the function

The correct answer is: The nature of critical points

The second derivative test is a method used in calculus to determine the nature of critical points of a function. Critical points occur where the first derivative of a function is either zero or undefined, potentially indicating local maxima, local minima, or points of inflection. The second derivative provides information about the concavity of the function at those critical points. Specifically, if the second derivative at a critical point is positive, the function is concave up, indicating that the point is a local minimum. Conversely, if the second derivative is negative, the function is concave down, suggesting that the point is a local maximum. If the second derivative equals zero, the test is inconclusive, and further analysis is needed. Understanding this testing method is crucial for optimizing functions in various contexts, such as physics, economics, and engineering, where determining the behavior of functions is essential. Thus, the utility of the second derivative test lies squarely in its ability to classify critical points based on the behavior of the function around them.