Example1:
Considerthe function :
The functionhas zerosat and has a relativemaximumat and a relativeminimum
at. Notethat the graphappearsto be concavedownfor all intervalsin and concave
up for all intervalsin. Wheredo you thinkthe concavityof the graphchangedfrom concavedown
to concaveup? If you answeredat you wouldbe correct.In general,we wishto identifyboth the
extremaof a functionand also points,the graphchangesconcavity. The followingdefinitionprovidesa formal
characterizationof suchpoints.
Definition: A pointon a graphof a function wherethe concavitychangesis calledaninflectionpoint
.
The exampleabovehad only one inflectionpoint.But we can easilycomeup with examplesof functions
wherethereare morethan one pointof inflection.
Example2:
Considerthe function
We can see that the graphhas two relativeminimums,one relativemaximum,and two inflectionpoints(as
indicatedby arrows).
In generalwe can use the followingtwo tests for concavityand determiningwherewe have relativemaximums,
minimums,and inflectionpoints.
Test for Concavity