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Calculus: Early Transcendentals
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.