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A function with this property is called strictly increasing (also increasing). [3] [4] Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing). [3] [4] A function with either property is called strictly monotone.
In other words, cubes (strictly) monotonically increase. Also, its codomain is the entire real line: the function x ↦ x 3 : R → R is a surjection (takes all possible values). Only three numbers are equal to their own cubes: −1, 0, and 1. If −1 < x < 0 or 1 < x, then x 3 > x. If x < −1 or 0 < x < 1, then x 3 < x.
In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions.
Monotone comparative statics is a sub-field of comparative statics that focuses on the conditions under which endogenous variables undergo monotone changes (that is, either increasing or decreasing) when there is a change in the exogenous parameters.
A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. [3] [4] Points where concavity changes (between concave and convex) are inflection points. [5]
Stated precisely, suppose that f is a real-valued function defined on some open interval containing the point x and suppose further that f is continuous at x. If there exists a positive number r > 0 such that f is weakly increasing on (x − r, x] and weakly decreasing on [x, x + r), then f has a local maximum at x.
His results were later generalized by Connell and Rasmussen, [3] who give necessary and sufficient conditions for concavifiability. They show that the function ( x , y ) ↦ f ( x , y ) = e e x ⋅ y {\displaystyle (x,y)\mapsto f(x,y)=e^{e^{x}}\cdot y} violates their conditions and thus is not concavifiable.
it is strictly increasing; it is s.t. () =. In fact, this is nothing but the definition of the norm except for the triangular inequality. Definition: a continuous function : [,) [,) is said to belong to class if: