Derivatives can describe a function in multiple ways.
Figure 5.2.1 Stationary points of a curve
Suppose is an interval in the domain of , so is defined for all in . Then, we have:
1.
is increasing on if for all in
2.
is decreasing on if for all in
3.
has a stationary point when :
a.
Local maximum (increases then decreases: derivative sign changes from + to -)
b.
Local minimum (decreases then increases: derivative sign changes from - to +)
c.
Stationary inflection (derivative sign does not change)
4.
is concave up on if for all in
5.
is concave down on if for all in
6.
has an (non-stationary) inflection point where
Figure 5.2.2 Example of a cubic function with a sign diagram on the right
Represent the sign changes in a sign diagram to identify each stationary point.
