CK12 Calculus - Single Variable

Example3:

Let’s examinethe function

1. Find the criticalvaluesfor which

or

at

Notethat when

2. Applythe First and SecondDerivativeTests to determineextremaand pointsof inflection.

We can note the signsof and in the intervalspartionedby

Key intervals Shapeof graph

Increasing,concavedown

Decreasing,concavedown

Decreasing,concaveup

Increasing,concaveup

Also note that By the SecondDerivativeTest we have a relativemaximumat
or the point

In addition, By the SecondDerivativeTest we havea relativeminimumat or the
point Now we can sketchthe graph.

LessonSummary

1. We learnedto identifyintervalswherea functionis concaveupwardor downward.


2. We appliedthe First and SecondDerivativeTests to determineconcavityand sketchgraphs.
   ReviewQuestions


3. Find all extremausingthe SecondDerivativeTest.