Discuss the similarities and the differences of the First Derivative Test and the Second Derivative Test for relative extrema.
Derivative Tests for Relative Extrema:
The derivatives of a function are closely related to the graph of a function, and there are rules for how the first and second derivatives can be used when analyzing a function for relative maximum and minimum points.
The first derivative test says that if a function is continuous at a relative extreme point then the first derivative at that point either equals zero or does not exist (e.g. a cusp)
The second derivative test says that if the second derivative is positive at a local extreme point then that point is a local minimum. Similarly if the second derivative is negative, the point is a relative maximum. If the second derivative is zero, or does not exist, then the second derivative test is inconclusive.
Answer and Explanation:
Both derivative tests are ways to relate the derivatives of a function to the shape of the graph and especially, to finding relative extrema.
The first derivative test is employed as a way to help find a relative extreme point whereas the second derivative test is a way to determine whether such a point is a max or min. So the distinction is between finding vs. classifying extrema.
A similarity is that neither test is fool-proof or guaranteed to give complete information. For example, while every extreme point is a critical point, not every critical point is a relative extreme point. Likewise, the second derivative test fails to give a complete final answer if the second derivative is either zero or does not exist.
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from AP Calculus AB: Exam PrepChapter 10 / Lesson 2
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