A sharp corner at a point on a graph indicates which of the following?

A sharp corner on a graph indicates that the function has a discontinuity in its slope at that specific point. Since a derivative represents the slope of a function at a point, if there is a sharp corner, the slope before the corner is different from the slope after the corner. This discrepancy means that the derivative does not exist at that point because you cannot define a single tangent line that accurately describes the behavior of the function both before and after the corner. Therefore, saying that the derivative is undefined is correct.

While other options may relate to characteristics of functions or points, they do not accurately represent the implication of a sharp corner in the context of calculus and derivatives. The concept of concavity is related to the curvature of a graph, and a local maximum could exist at a corner, but does not necessarily pertain to the defining feature of a sharp corner itself—the undefined nature of the derivative.