Sketching the Graph of a Function

In this topic:

    1. Review the First Derivative Test, Concavity Test and the Second Derivative Test for the functions: and
    

In this topic we give some examples that further illustrate the uses of the First Derivative Test, the Concavity Test, and the Second Derivative Test.

Example (Sketching the Graph of a Function)  For the function . Find all first and second order critical numbers. Apply the First Derivative Test, Concavity Test, and the Second Derivative Test. Sketch the graph of the function.

    Solution. The first and second derivatives are:





Solving we find the first order critical numbers to be Solving we find the second order critical numbers to be   Applying the First Derivative Test we find,  



Checking these local extrema by using the Second Derivative Test: determining and so is a local minimum and and so is a local maximum. Applying the Concavity Test:  



A sketch of the graph of follows:

Example (Sketching the Graph of a Function)  For the function . Find all first and second order critical numbers. Apply the First Derivative Test, Concavity Test, and the Second Derivative Test. Sketch the graph of the function.

    Solution. The first and second derivatives are:





Note that is not a critical number because is not defined. Solving we find the first order critical numbers to be Solving we find the second order critical numbers to be   Applying the First Derivative Test we find,  



Checking this local extrema by using the Second Derivative Test: determining



and so

is a local minimum.

Applying the Concavity Test:  



A sketch of the graph of follows:

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Sketching The Graph Of A Function
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/sketching-the-graph-of-a-function.html