Wetzel

Math 140

3.4 – Concavity and the Second Derivative Test

Concavity –

            Concave Up –   is increasing

            Concave Down –   is decreasing

Test for Concavity –

            Let  be a function whose second derivative exists on an open interval .

1.      If  for all x in , then the graph of  is concave upward in .

2.      If  for all x in , then the graph of  is concave downward in .

Points of Inflection –

            where the graph of a function changes from downward to upward (or upward to downward)

            X-values where  or wheredoes not exist are possible points of inflection.

Second Derivative Sign Diagram –

Ex.:

1.)                                                             2.) 

3.) 

HW:  p.195 #1, 3, 5, 11-21odd (construct a 2^nd derivative sign diagram)