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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]
This follows from the fact that the convolution of two log-concave functions is log-concave. The product of two log-concave functions is log-concave. This means that joint densities formed by multiplying two probability densities (e.g. the normal-gamma distribution, which always has a shape parameter ≥ 1) will be log-concave.
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Specifically, a twice-differentiable function f is concave up if ″ > and concave down if ″ <. Note that if f ( x ) = x 4 {\displaystyle f(x)=x^{4}} , then x = 0 {\displaystyle x=0} has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is ...
A sigmoid function is a bounded, differentiable, ... and it is concave for values greater than that point: in many of the examples here, that point is 0.
In simple terms, a convex function graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap . A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. [1]
First, it’s important to understand that inflammation isn’t always bad. “Inflammation is one of the body’s key mechanisms for maintaining homeostasis, acting as a natural response to ...
As can be seen, this function is not convex because of the concavity, and it is not pseudoconvex because it is not differentiable at =. Quasiconvex function that is not convex, nor pseudoconvex. Generalization to nondifferentiable functions