"

Corollary 2 — On the Sign of the Derivative

Under the conditions of the theorem, the derivative of f is either positive
throughout the interior of I or negative throughout the interior of I.

Proof

By assumption, the derivative exists and does not take the value 0 in the interior of I.  The function is strictly monotone over I.  If it is increasing, the derivative cannot be
negative, so it is everywhere positive in the interior of I.  If the function is decreasing, the derivative cannot be positive, so it is everywhere negative in the interior of I.

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A Slightly Stronger Result for Monotone Functions Copyright © 2022 by Larry Lipskie is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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