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Problem 93 Hard Difficulty

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 functions
but their derivatives change sign infinitely often on both
sides of $0 .$
(b) Show that $f$ hat $f$ has neither a local maximum nor a local
minimum at $0, g$ has a local minimum, and $h$ has a local
maximum.

Answer

a) See explanation
b)

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