Calculus Methods

07 Finding Absolute Extrema

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        Absolute extrema occur either at critical points or at

        endpoints.

 

         1) Find all critical numbers.

 

                  A) Take the derivative of the function.

 

                  B) Express the derivative as a single fraction.

 

                  C) Critical points occur when the derivative is

                    equal to zero or when the derivative has a

                    zero in the denominator, so set both the

                    numerator and denominator equal to zero.

 

                  D) Solve for the critical numbers.

 

        2) Evaluate the function at each critical number and

        at each endpoint.

 

        3) The largest value is the maximum value of the

        function and the smallest value is the minimum

        value of the function.

 

        4) Make sure to answer the question! If it asks for

        the x-position of each extremum, only list the

        x-values. If it asks for the function values of each

        extremum, list only the function (y) values. If the

        question asks for "the maximum" or "the

        minimum," then the question is asking for the

        coordinates.

 

 

 

        Example #1: Find the maximum of the function

        on the interval .

 

        1A)

 

 

        1B) There is no fraction in this derivative.

 

 

        1C)

 

 

        1D)

 

 

        2)

 

        

        3)  So the x-values at which the maxima occur are

        and . The maximum value is 2. The maxima

        are and .

 

 

        4) The answer to the question is that the maximum

        value of the function is 2.

 

 

 

 

        Example #2: Find the extrema of the function

        on the interval .

 

 

        1A)

 

 

 

        1B)

 

 

 

        1C)

 

        1D)

 

        2) Note that we can ignore and since

        those are outside of the interval at which we are

        looking.

                 

 

 

        3) The maximum value on the interval is . The

        x-position at which this maximum occurs is. The

        maximum is . The minimum value is . The x-

        position at which this minimum occurs is . The

        minimum is .

 

        4) The answer to the question is that there is a

        maximum at and a minimum at .