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The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions $ f $, $ g $, and $ h $ whos values at $ 0 $ are all $ 0 $ and, for $ x \not= 0 $,$$ f(x) = x^4 \sin \frac{1}{x} $$ $$ g(x) = x^4 \left(2 + \sin \frac{1}{x} \right) $$$$ h(x) = x^4 \left(-2 + \sin \frac{1}{x} \right) $$(a) Show that $ 0 $ is a critical number of all three functions but their derivatives change sign infinitely often on both sides of $ 0 $.(b) Show that $ f $ has neither a local maximum nor a local minimum at $ 0 $, $ g $ has a local minimum, and $ h $ has a local maximum.

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Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Derivatives

Differentiation

Volume

Harvey Mudd College

Baylor University

University of Nottingham

Idaho State University

Lectures

04:35

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

06:14

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

19:39

The three cases in the Fir…

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12:19

01:14

01:06

08:39

The Second Derivative Test…

02:57

The (second) second deriva…

03:48

01:45

01:01

The first derivative $f^{\…

01:18

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