Calculus Methods

08 Finding Relative Extrema

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         Relative extrema occur either at critical points or at

        endpoints, but a typical AP problem will not ask

        about relative extrema and endpoints in the same

        problem.

 

        Relative extrema are more difficult to identify,

        because we must know what the slope does on

        either side of each critical point.

 

        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. This step is necessary since we do

        not want to test a critical number if it doesn't

        correspond to an actual point!

 

 

        3) If you are not dealing with a fraction, use the

        second derivative test:

 

                  A) Find the second derivative of the function.

 

                  B) Express the second derivative as a single

                    fraction.

 

                  C) Evaluate the second derivative at each x-

                    value of each critical point. If the second

                    derivative is less than zero, that critical point

                    is a maximum (negative concavity). If the

                    second derivative is greater than zero, that

                    critical point is a minimum (positive

                    concavity). If the second derivative is zero

                    or undefined, the second derivative test fails

                    and you must move on to step 4.

 

 

        4) You will only use this step if the second

        derivative test fails or if the second derivative is

        difficult to find. The other way to determine if a

        point is a critical point is by seeing how the slope

        changes near that critical point. If the slope changes

        from positive to negative, the critical point must be

        a maximum. If the slope changes from negative to

        positive, the critical point must be a minimum. If the

        slope is unchanged, the critical point must be a

        point of inflection.

                  Since we need to know how the sign of the

        slope changes, we want to observe when the slope

        is positive and when it is negative. As we saw in

        method 02, one way to do this is with a sign chart.

        Make the sign chart OF THE FIRST DERIVATIVE

        to determine where it is positive and where it is

        negative.

 

 

        5) If you need to determine when an endpoint is a

        maximum or a minimum, evaluate the FIRST

        derivative at each endpoint. For a beginning

        endpoint, if the slope is positive, then the function is

        headed up from the beginning and the beginning

        endpoint must be a relative minimum. Similarly, if

        the slope is negative, the beginning endpoint must

        be a relative maximum. For the ending endpoint, the

        opposite is true: positive slope indicates a maximum

        and negative slope indicates a minimum.

 

 

 

        Example #1: Find all relative extrema of the

        function .

 

        1A)

 

 

        1B) There is no fraction in this derivative.

 

 

        1C)

 

 

        1D)

 

 

        2)

        So both of the critical numbers are critical points.

 

 

        3A)

 

        3B) The second derivative is not a fraction.

 

        3C)

 

 

        So there is a relative maximum at and there is a

        relative minimum at . Note that we have not

        determined whether these extrema are also

        absolute. To determine that, see method 07.

 

 

 

         Example #2: Find all relative extrema of the

        function .

 

        1A)

 

        1B) There is no fraction in this derivative.

 

        1C)

 

        1D)

 

 

        2)

 

         So both of the critical numbers are critical points.

 

 

        3) Let's pretend that the second derivative test

        failed so that we must use step 4.

 

 

        4) We will set up a sign chart for the derivative to

        see how the slope changes around the critical points

        (see method 02 for instructions on how to set up a

        sign chart).

 

 

 

 

 

 

 

 

 

        Around the point when , the sign changes from

        negative to positive, so the point is a

        minimum. Around the point when , the sign

        changes from positive to negative, so the point

        is a maximum.

 

 

        

On to Method 09 - Finding "Other" Points Of Inflection

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