Graphs of Functions & Their Derivatives

Sarah Miller

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Study Guide Overview

This study guide covers the Extreme Value Theorem, focusing on its application to continuous functions over closed intervals. It explains how to identify minimum and maximum values, including their potential locations at endpoints or as local extrema. The guide includes a worked example, practice questions, and a glossary of key terms like continuous, differentiable, and local maximum/minimum. It also emphasizes the importance of verifying continuity and differentiability before applying the theorem.

#Extreme Value Theorem Study Notes

#Extreme Value Theorem

#What is the Extreme Value Theorem?

The extreme value theorem states that:

If a function $f$ is continuous over the closed interval $[a, b]$ , then:
- $f$ has at least one minimum value and at least one maximum value on the interval $[a, b]$ .

Key Concept

A continuous function will always have a minimum and a maximum value on a closed interval.

These values might occur at the endpoints of the interval. If a minimum or maximum value does not occur at an endpoint, it will be a local minimum or local maximum within the open interval.
- In this case, if the function is differentiable on $(a, b)$ , then the derivative of the function will be zero at that point.

#Exam...

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Question 1 of 6

If a function $f(x)$ is continuous on a closed interval $[a, b]$ , what does the Extreme Value Theorem guarantee about $f(x)$ on that interval? 🤔

It has at least one minimum value

It has at least one maximum value

It has at least one minimum and at least one maximum value

It has a derivative of zero at some point