Defininition: If the graph of f(x) lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on an interval I, then it is called concave downward on I.

Test for Concavity: If f is twice differentiable on an interval I then

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

Defininition: A point P on a curve is called a point of inflection if the curve changes concavity (upwards to downwards or vice versa.)

The Second Derivative Test: If f'' is continuous on an open interval that contains c:

1. If f'(c) = 0, and f''(c) > 0, f has a local min at c.
2. If f'(c) = 0, and f''(c) < 0, f has a local max at c.

Last modified Wed Nov 6 23:04:49 CST 1996