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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 $h$ whose values at 0 are all 0 and, for $x \neq 0$ ,$$f(x)=x^{4} \sin \frac{1}{x} \quad g(x)=x^{4}\left(2+\sin \frac{1}{x}\right)$$$$h(x)=x^{4}\left(-2+\sin \frac{1}{x}\right)$$(a) Show that 0 is a critical number of all three functionsbut their derivatives change sign infinitely often on bothsides of $0 .$(b) Show that $f$ hat $f$ has neither a local maximum nor a localminimum at $0, g$ has a local minimum, and $h$ has a localmaximum.

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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

Oregon State University

Baylor University

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…

00:21

Show that $f(x, y)=x^{2}$ …

01:05

Consider the functions $f(…

02:28

01:46

Derivative Test Locate the…

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