The FirstDerivativeTest
LearningObjectives
A studentwill be able to:
- Find intervalswherea functionis increasingand decreasing.
- Applythe First DerivativeTest to find extremaand sketchgraphs.
Introduction
In this lessonwe will discussincreasingand decreasingpropertiesof functions,and introducea method
with whichto studythesephenomena,the First DerivativeTest. This methodwill enableus to identifyprecisely
the intervalswherea functionis eitherincreasingor decreasing,and also help us to sketchthe graph.Note
on notation:The symbolε and are equivalentand denotethat a particularelementis containedwithina
particularset.
Definition: A function is said to beincreasingon containedin the domainof if
whenever for all A function is said to be decreasingon containedin
the domainof if whenever for all
If whenever for all then we say that is strictlyincreasing
on If whenever for all then we say that is strictly
decreasingon
We saw severalexamplesin the Lessonon Extremeand the MeanValue Theoremof functionsthat had
theseproperties.
Example1:
The function is strictlyincreasingon :
Example2: