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On l’Hospitaltype rules for monotonicity
 J. Inequal. Pure Appl. Math
"... 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 li ..."
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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 ˜ρ: = r ′ · g 2 /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.
“NONSTRICT” L’HOSPITALTYPE RULES FOR MONOTONICITY: INTERVALS OF CONSTANCY
, 2008
"... Let f and g be differentiable functions defined on the interval (a,b), where − ∞ � a < b � ∞, and let r: = f g and ρ:= It is assumed throughout that g and g ′ do not take on the zero value anywhere on (a,b). The function ρ may be referred to as a derivative ratio for the “original ” ratio r. In ..."
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Let f and g be differentiable functions defined on the interval (a,b), where − ∞ � a < b � ∞, and let r: = f g and ρ:= It is assumed throughout that g and g ′ do not take on the zero value anywhere on (a,b). The function ρ may be referred to as a derivative ratio for the “original ” ratio r. In [11], general “rules ” for monotonicity patterns, resembling the usual l’Hospital rules for limits, were given. In particular, according to [11, Proposition 1.9 and Remark 1.14], one has the dependence of the monotonicity pattern of r ( on (a,b)) on that of ρ (and also on the sign of gg ′ ) as given by Table 1. The vertical double line in the table separates the conditions (on the left) from the corresponding conclusions (on the right). ρ gg ′ r> 0
L'HospitalType Rules for Monotonicity: Include Them into Calculus Texts!
"... Ratios are ubiquitous. A recent Google search for \ratio " returned over 96 106 items. Among those are numerous and commonly used nancial ratios (visit, for example, ..."
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Ratios are ubiquitous. A recent Google search for \ratio " returned over 96 106 items. Among those are numerous and commonly used nancial ratios (visit, for example,