Concavity

Concavity refers to the curvature of the graph of a function. A graph is said to be concave up when the rate at which the values of the function increases is itself increasing, and said to be concave down when the this rate is decreasing. Graphs which are concave up often look similar to a U shape, and vice-versa when they are concave down.

To formally determine the concavity of a graph, one must use that graph's second derivative. If the second derivative at a point on the graph is negative, then the graph at that point is concave down, and vice-versa should it be positive. If the second derivative at a point equals zero, then there is no concavity; should the concavity be different on either side of that point, then this point is said to be an inflection point.

Concavity can also be used to determine the maxima and minima of functions by means of the Second Derivative Test. Simply put, one can apply the second derivative to determine the concavity of critical points, or points where a maximum or minimum can occur. If the point is concave down, then the point must be a maximum, and if it is concave up, it must be a minimum.