#First Derivative Test

#Table of Contents

1. Introduction
2. Relationship Between First Derivative and Local Extrema
3. First Derivative Test
4. Worked Example
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

#Introduction

Understanding how the first derivative of a function is related to its local extrema (maximums and minimums) is crucial in calculus. This study note will explore the first derivative test, a method used to identify and classify these extrema.

#Relationship Between First Derivative and Local Extrema

#Explanation

Local extrema are found at critical points where the first derivative, $f'(x)$, is equal to zero. However, not all points where $f'(x) = 0$ are local extrema. For example, the graph of $y = x^3$ has a derivative of zero at $x = 0$, but this point is not a minimum or maximum; it is a point of inflection.

A more precise definition for local minimums and maximums is:

• If $x = a$ is a critical point of $f(x)$ (i.e., $f'(a) = 0$) and if:
  • $f'(x)$ changes...