Which derivative is used to determine the nature of stationary points?

To determine the nature of stationary points, the second derivative is utilized. When analyzing a function, stationary points occur where the first derivative equals zero, indicating potential maximum, minimum, or inflection points. However, to classify these points, the second derivative comes into play.

The second derivative provides information about the concavity of the function at the stationary points. If the second derivative is positive at a stationary point, it indicates that the graph of the function is concave up at that point, implying a local minimum. Conversely, if the second derivative is negative, the graph is concave down, indicating a local maximum. If the second derivative equals zero, the test is inconclusive, suggesting further analysis may be required to ascertain the nature of the stationary point.

Thus, the effective use of the second derivative test is essential for identifying whether stationary points represent local minima, local maxima, or neither.