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Graphs of Functions & Their Derivatives

Sarah Miller

Sarah Miller

7 min read

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Study Guide Overview

This study guide covers local and global extrema, focusing on the identification and classification of critical points. It explains the different types of critical points: local minimums and maximums, points of inflection, and points where the derivative is undefined. Examples of finding critical points for polynomial and root functions are provided, along with practice questions and exam strategies.

Study Notes: Local Extrema versus Global Extrema

Table of Contents

  1. Introduction
  2. Local Extrema vs. Global Extrema
  3. Critical Points
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways
  7. Exam Strategy

Introduction

In calculus, understanding the concepts of local and global extrema and critical points is essential for analyzing and interpreting the behavior of functions. This guide will help you distinguish between these concepts, identify critical points, and apply this knowledge effectively in exams.

Local Extrema vs. Global Extrema

Definitions

  • Extrema: Refers to the maximum and minimum points on the graph of a function.
  • Global (Absolute) Extrema: The highest (maximum) or lowest (minimum) value of the function over its entire domain.
  • Local (Relative) Extrema: The highest or lowest value of the function within a specific interval of the domain.
Key Concept

Global extrema are the absolute highest or lowest points of a function over its entire domain, while local extrema are the highest or lowest points within a specific interval.

Consider the function f(x)=x33x2+4f(x) = x^3 - 3x^2 + 4: - The global minimum occurs at x=x = -\infty. - The local minima and maxima occur at points where the function's derivative equals zero.
Exam Tip

Remember that every global...

Question 1 of 10

What's the difference between a local and a global maximum? 🤔

A local maximum is always the highest point on the entire graph

A global maximum is the highest point within a specific interval

A global maximum is the absolute highest point on the entire graph, while a local maximum is a peak within a specific interval

They are the same thing