Ohio Assessments for Educators (OAE) Mathematics Practice Exam

When the second derivative is less than zero, what does it indicate?

1. Minimum

2. Maximum

3. Neither

4. Both

The correct answer is: Maximum

When the second derivative of a function is less than zero, it indicates that the function is concave down at that point. This concavity suggests that the function is experiencing a local maximum. In mathematical terms, if the first derivative is zero (indicating a critical point) and the second derivative is negative, this confirms that the critical point is a local maximum. The reasoning behind this is grounded in the behavior of the tangent line and the shape of the graph. If the graph is concave down, any slight movement away from the critical point results in lower function values, further reinforcing that the critical point represents a peak in the curve. Understanding the role of the second derivative in determining concavity is crucial, as it gives insight into the nature of stationary points and helps in analyzing the function's overall behavior regarding local extrema.