Second Derivative Test

The Second Derivative Test is a method in calculus used to determine whether a critical point of a function is a local maximum, local minimum, or a saddle point. The test involves analyzing the value of the second derivative of the function at the critical point.

If a function f(x) has a critical point at x = c where f'(c) = 0, the Second Derivative Test states:

1. If f”(c) > 0, then the function has a local minimum at x = c.
2. If f”(c) < 0, then the function has a local maximum at x = c.
3. If f”(c) = 0, the test is inconclusive, and further analysis is needed to determine the nature of the critical point.

The Second Derivative Test provides an alternative to the First Derivative Test when determining the nature of a critical point. It is particularly useful when the sign of the first derivative does not change at the critical point or when it is difficult to evaluate the sign of the first derivative near the critical point.