CK12 Calculus - Single Variable

Supposethat is continuouson and that is someopenintervalin the domainof

1. If for all then the graphof is concaveupwardon

2.If for all then the graphof is concavedownwardon

A consequenceof this concavitytest is the followingtest to identifyextremevaluesof
SecondDerivativeTest for Extrema

Supposethat is a continuousfunctionnear and that is a criticalvalueof Then

1. If then has a relativemaximumat


2. If then has a relativeminimumat


3. If then the test is inconclusiveand may be a pointof inflection.

Recallthe graph We observedthat and that therewas neithera maximumnor minimum.
The SecondDerivativeTest cautionsus that this may be the casesinceat at
So now we wish to use all that we havelearnedfrom the First and SecondDerivativeTests to sketchgraphs
of functions.The followingtableprovidesa summaryof the testsand can be a usefulguidein sketching
graphs.

Information from applying First and SecondShapeof the graphs

DerivativeTests

Signsof first and secondderivatives

is increasing

is concaveupward

is increasing

is concavedownward

is decreasing

is concaveupward

is decreasing

is concavedownward

Lets’ look at an examplewherewe can use both the First and SecondDerivativeTests to find out information
that will enableus to sketchthe graph.