corollary
corollary is a statement that follows directly from a theorem.
Definition
corollary is a statement that follows directly from a theorem. In Theorem 4.7, Corollary 2 establishes that if two functions f and g are differentiable over an interval J and their derivatives are equal for all x in J, then f(x) equals g(x) plus a constant C. The proof involves defining h(x) as f(x) - g(x) and applying Corollary 3 to show that f' is decreasing over J. The theorem also notes that if f'(a) = 0, then for all x in J, f'(x) > 0 when x ≠ a.
Mechanism
Theorem 6.16 establishes corollary as a derivative consequence of the natural logarithm's differentiability. This corollary confirms that In x is continuous, following from its differentiability. Theorem 4.8's corollary 3 relates to increasing/decreasing functions, linking continuity and differentiability over intervals.
Causes
corollary The existence of a differentiable function f on an interval J implies the Mean <a href='/en/entity/value-theorem'>Value Theorem</a> applies, ensuring there exists c ∈ J where f'(c) ≠ 0. This contradicts the assumption that f'(x) = 0 for all x ∈ J. From Corollary 1, functions with identical derivatives differ by at most a constant.
Effects
corollary The corollary establishes that functions with identical derivatives differ by at most a constant. This follows from the Mean <a href='/en/entity/value-theorem'>Value Theorem</a>, which guarantees the existence of points where the derivative is non-zero. Such points contradict the assumption of a zero derivative across an interval. The corollary also links to derivative properties, showing how differentiable functions behave under specific conditions. These effects highlight the relationship between derivatives and function behavior.
Decreasing Function
corollary By Corollary 3, the derivative f' is a decreasing function over J. Since f'(a) = 0, this implies that for all x in J, f'(x) > 0 if x < a. The corollary establishes that the function's derivative maintains a decreasing trend across the interval J.
Since Isa Causes
corollary Since isa causes are rooted in the derivative's role in differentiable functions, the Mean <a href='/en/entity/value-theorem'>Value Theorem</a> ensures the existence of c € (a, b) where f'(c) ≠ 0. This contradicts the assumption that f'(x) = 0 for all x € J, as shown by Corollary 1. The corollary also states that functions with identical derivatives differ by at most a constant, reinforcing the relationship between derivative zero conditions and function behavior.
Theorem Derivative Mechanism
corollary serves as a derivative mechanism within the theorem derivative framework, specifically tied to the natural logarithm's properties. Theorem 6.16 establishes that the function In x is differentiable, which directly supports the continuity of the natural logarithm. This corollary reinforces the relationship between differentiability and continuity in the context of the natural logarithm's derivative. As a derivative mechanism, it provides a foundational link between the theorem's core assertion and its broader implications. The evidence underscores how the corollary operates within the theorem derivative structure to validate the natural logarithm's continuous behavior.