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Derivative Sign Patterns

by Jeffrey Clark

This article originally appeared in:
College Mathematics Journal
November, 2011

Subject classification(s): Calculus | Single Variable Calculus | Differentiation
Applicable Course(s): 3.2 Mainstream Calculus II

Motivated by the observation that the derivatives of \(e^x\) are all positive and the derivatives of \(e^{-x}\) alternate sign, the author asks what kinds of ``sign patterns" are possible for infinitely differentiable real functions. The author shows that infinitely differentiable functions with domain \(\mathbb{R}\) can exhibit one of only four possible sign patterns.  However, Taylor series are used to show that all sign patterns are possible if the domain is a bounded open interval.

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Capsule Course Topic(s):
One-Variable Calculus | Differentiation: Definition and Elementary Application
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