The tutor explains concavity and point of inflection with an example.

Concavity refers to an aspect of graph shape. My first-year calculus professor explained it this way: concave upward will collect rain, while concave downward will shed rain. Numerically, when the second derivative is positive, the graph is concave upward. When the second derivative is negative, the graph is concave downward.

In the graph above, the section from P to Q is concave downward; from Q to R is concave upward.

A point where concavity changes from negative to positive (or positive to negative) is called a point of inflection. In the graph above, Points Q and R are inflection points.

At a point of inflection, the second derivative is either 0 or undefined. However, f”=0 doesn’t guarantee a point of inflection; you still have to check either side (if you can’t see the graph).

The graph above, y=sinx, has second derivative -sinx. At point Q (where x=Π), -sin(Π)=0. Just to the left, -sin3=-0.1411. (Recall Π=3.14159….) Past Π, -sin3.3=0.1577. The sign change across Π confirms the inflection point at (Π,0).

Source:

Larson, Roland and Robert Hostetler. Calculus, part one. Toronto: D C Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.