What does a zero second derivative indicate about a stationary point?

A zero second derivative at a stationary point indicates that the concavity of the function changes, which characterizes a point of inflection. At a stationary point, the first derivative equals zero, suggesting that the slope of the curve is flat (horizontal tangent). The second derivative provides insight into the curvature of the function at that point. If the second derivative is zero, it does not definitively indicate a local maximum or minimum since those conditions are characterized by a positive or negative second derivative, respectively. Instead, it suggests that the graph may change its concavity at that point, meaning the function transitions from concave up to concave down or vice versa. This transition marks it as a point of inflection, where the nature of the function can switch, aligned with the characteristic of having a zero second derivative.