Concavity, Convexity and Points of Inflection

If the graph of flies above all of its tangents on an interval I, then it is called concave upward (convex downward) on I. If the graph of flies below all of its tangents on I, it is called concave downward (convex upward) on I.

Second Derivative Test

The second derivative helps to determine the intervals of concavity or convexity.

Suppose f is twice differentiable on an interval I.

  1. If f′′(x) > 0 for all x in I, then the graph of f is concave upward (convex downward) on I.
  2. If f′′(x) < 0 for all x in I, then the graph of f is concave downward (convex upward) on I.

Inflection Point

A point P on a curve is called a point of inflection if the curve changes from concave upward (convex downward) to concave downward (convex upward) or from concave downward (convex upward) to concave upward (convex downward) at P. The point that separates the convex part of a continuous curve from the concave part is called the point of inflection of the curve.

At the point of inflection the tangent line, if it exists, cuts the curve, because on one side the curve lies under the tangent and on the other side, above it.

Let a curve be defined by an equation y = f(x). If f′′(x0) = 0 or f′′(x0) does not exist and if the derivative f′′(x) changes sign when passing through x = x0, then the point of the curve with abcissa x = x0 is the point of inflection.