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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
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The three cases in the Fir…
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The Second Derivative Test…
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