First Derivative Test

Proposition (First Derivative Test) Suppose that is a critical number of a function that is continuous on Then the following statements hold:

    (i) If for and for then has a relative (local) minimum at

    (ii) If for and for then has a relative (local) maximum at

    (iii) If neither (i) nor (ii) hold then has no relative (local) extremum at
    
Example (First Derivative Test) Apply the First Derivative Test to find the local extrema of the function and sketch its graph.

    Solution. First we find the critical numbers of by solving and determining where is undefined but is defined. We find,
    


Solving we find Also does not exist but and therefore the critical numbers are and We determine the local extrema using the following table:



Therefore, is a local maximum and is a local minimum. Here is the graph of the function



Notice there is a corner at because is defined there but is not.

Example (First Derivative Test) Find the local and absolute extrema values of the function on the interval Sketch the graph.

    Solution. First we find the critical numbers of by solving and determining where is undefined but is defined. We find,
    


To find the critical numbers we set and obtain We determine the local extrema and absolute extrema using the following table:



The function does not have a local extrema at The local maximum is and the local minimum is To determine absolute extrema we compute the functional values at the endpoints, namely and Therefore, the absolute maximum is and the absolute minimum is  

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First Derivative Test
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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