Question

The function has continuous second derivatives, and a critical point at suppose. Then the point:

1) Cannot be determined
2) Is a local maximum
3) Is a local minimum
4) Is an inflection point

Answer 1

If the information about the second derivative is not given, we cannot determine the nature of the critical point.

Step-by-step explanation:

The point can be determined based on the given information. If the function has continuous second derivatives and a critical point at x = a, we can analyze the behavior of the second derivative at x = a to determine the nature of the critical point.

1. If the second derivative is negative at x = a, then the critical point is a local maximum.
2. If the second derivative is positive at x = a, then the critical point is a local minimum.
3. If the second derivative is zero at x = a, then further analysis is needed to determine the nature of the critical point (inflection point).

In this case, since the information about the second derivative at x = a is not given, we cannot determine the nature of the critical point.