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Sketching polynomial functions
Using the first derivative
Using the second derivative
Using the original function
Example: x^3 – 6x^2 + 1
First, find the formulas for the first and second derivatives.
Next, analyze the first derivative. Compare your results to the ones you see here.
If you have forgotten how to analyze the first derivative, refer to Lesson 3 in Module 8.2.
In fact, complete details for this function can be found in Example 1 for that lesson.
Find all possible relative extrema:
Now, set up an x-axis that is divided into three pieces by the numbers 0 and 4
and use a test value in each piece.

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If you have forgotten how to analyze the second derivative, refer to Lesson 3 in Module 8.3.
Again, complete details for this function can be found in Example 1 for that lesson.
Find all possible inflection points:
Now, set up an x-axis that is divided into two pieces by the number 2
and use a test value in each piece.

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