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If a real-valued, differentiable function f, defined on an interval I of the real line, has zero derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity and one-sided differentiability of f. The version for right differentiable functions is ...
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. [1] As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in ...
The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point x {\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows:
Even Markowitz, himself, stated that "semi-variance is the more plausible measure of risk" than his mean-variance theory. [5] Later in 1970, several focus groups were performed where executives from eight industries were asked about their definition of risk resulting in semi-variance being a better indicator than ordinary variance. [6]
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
Rigorously, a subderivative of a convex function : at a point in the open interval is a real number such that () for all .By the converse of the mean value theorem, the set of subderivatives at for a convex function is a nonempty closed interval [,], where and are the one-sided limits = (), = + ().
Semiderivative or Semi-derivative may refer to: One-sided derivative of semi-differentiable functions Half-derivative , an operator H {\displaystyle H} that when acting twice on a function f {\displaystyle f} gives the derivative of f {\displaystyle f} .
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the L p space ([,]).