Second Derivatives and the Shape of a Graph

A graph's concavity—its upward or downward bend—comes from its function’s second derivative, which shows how the graph curves and how the slope changes.

A positive second derivative means the graph is concave up. The slope increases in these regions, and the graph lies above its tangent lines.

A negative second derivative means the graph is concave down. In these regions, the slope decreases, and the graph lies below its tangent lines.

If the second derivative is zero or undefined, the point may be an inflection point. At this point, the graph changes from concave up to down, or vice versa.

These points divide the domain into intervals for testing concavity. The second derivative test is used at critical points found from the first derivative.

If the second derivative is positive at a critical point, the graph curves upward and the point is a local minimum.

If the second derivative is negative at a critical point, the graph curves downward, and the point is a local maximum.

In marketing, the second derivative shows how returns change. A concave-down ad-benefit graph means extra spending gives smaller gains, while a concave-up curve means gains grow faster.