Second Derivative Test

In this topic:

    1. Define First Order and Second Order Critical Points
    
    2. State the Second Derivative Test
    
    3. Illustrate with the functions and
    

    Another application of the second derivative is the second derivative test.

Definition (Critical Points) We will call the number a first order critical number if or does not exist and a second-order critical number if or does not exist.

Proposition (Second Derivative Test) Suppose in continuous on an open interval that contains with Then

    (i) If then has a relative (local) minimum at
    
    (ii) If then has a relative (local) maximum at

    Warning. The second derivative test doe not tell us anything if both and For example, if and both

and

The point is a minimum for but is neither a maximum not a minimum for However, the first derivative test is still useful.

Example (Second Derivative Test) Use the second derivative test to determine whether each critical number of the function corresponds to a relative maximum, a relative minimum, or neither.

    Solution. We find the first order critical numbers by



and so and are the critical numbers. Now to apply the Second Derivative Test we find the second derivative, Since the point is a local maximum and since the point is a local minimum.

Example (Second Derivative Test) Use the second derivative test to determine whether each critical number of the function corresponds to a relative maximum, a relative minimum, or neither.

    Solution. We find the first order critical numbers by



and so are the first order critical numbers of Note that, even though is undefined, so is and so is not a critical number. We compute and since the point is a local minimum and since the point is a local maximum.

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Second Derivative Test
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/second-derivative-test.html