Graphs of Functions & Their Derivatives
Sarah Miller
6 min read
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Study Guide Overview
This study guide covers the Second Derivative Test for classifying local extrema (minima and maxima). It explains how to use the first and second derivatives to identify and classify critical points, including cases where the second derivative is zero. The guide also discusses points of inflection and the first derivative test. Examples and practice questions are provided to reinforce these concepts.
#Second Derivative Test
#Table of Contents
- First and Second Derivatives at Key Points
- What is the Second Derivative Test?
- Points with a Second Derivative of Zero
- Worked Examples
- Practice Questions
- Glossary
- Summary and Key Takeaways
#First and Second Derivatives at Key Points
To be able to classify key points on the graph of a function, it is important that you are confident with the properties of the first and second derivatives at these points.
| Type of Point | First Derivative | Second Derivative |
|---|---|---|
| Local minimum | Zero | Positive or zero |
| Local maximum | Zero | Negative or zero |
| Point of inflection (critical) | Zero | Zero |
| Point of inflection (non-critical) | Non-zero | Zero |
It is essential to understand the behavior of first and second derivatives at these key points for effective classification.
#What is the Second Derivative Test?
The second derivative can be used to determine if a critical point is a local minimum or maximum.
The second derivative test states that:
- If and , then has a local minimum at .
- If and , then has a local **m...
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