On L’hospital-type Rules for Monotonicity

Abstract

Elsewhere we developed rules for the monotonicity pattern of the ratio r := f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ := f ′/g′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for limits, since in general we do not require that both f and g, or either of them, tend to 0 or ∞ at an endpoint or any other point of (a, b). Here new insight into the nature of the rules for monotonicity is provided by a key lemma, which implies that, if ρ is monotonic, then ρ̃ := r′ · g2/|g′| is so; hence, r′ changes sign at most once. Based on the key lemma, a number of new rules are given. One of them is as follows: Suppose that f(a+) = g(a+) = 0; suppose also that ρ↗↘ on (a, b) – that is, for some c ∈ (a, b), ρ↗ (ρ is increasing) on (a, c) and ρ↘ on (c, b). Then r ↗ or ↗↘ on (a, b). Various applications and illustrations are given.

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Cite this paper

@inproceedings{Pinelis2006OnLR, title={On L’hospital-type Rules for Monotonicity}, author={Iosif Pinelis}, year={2006} }