Graphs of Functions and Derivatives 115
concave
downward
f ′′< 0 f′′< 0
Figure 7.2-14
Figure 7.2-15
Points of Inflection
A point P on a curve is a point of inflection if:
- the curve has a tangent line at P, and
- the curve changes concavity at P (from concave upward to downward or from concave
downward to upward).
(See Figures 7.2-16–7.2-18.)
f′′< 0
f′′> 0
pt. of inflection
Figure 7.2-16
f′′< 0
f ′′> 0
pt. of inflection
Figure 7.2-17
f′′> 0 f′′< 0
CUSP
not a pt. of
inflection
Figure 7.2-18