5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 17:21

Graphs of Functions and Derivatives 149

concave

downward

f′′ < 0 f′′ < 0

Figure 8.2-14

concave

upward

f′′ > 0 f′′ > 0

Figure 8.2-15

Points of Inflection
A point P on a curve is a point of inflection if:

1. the curve has a tangent line at P, and


2. the curve changes concavity at P (from concave upward to downward or from concave
   downward to upward).

(See Figures 8.2-16---8.2-18.)

f′′ < 0

f′′ > 0

pt. of inflection

Figure 8.2-16

f′′ < 0

f′′ > 0

pt. of inflection

Figure 8.2-17

f′′ > 0 f′′ < 0

Cusp

not a pt. of

inflection

Figure 8.2-18