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The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly 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.
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x 2 /2 is a concave function of x. But f is not concave since the second derivative is positive for | x | > 1:
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 ]
In mathematics, concavification is the process of converting a non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It is especially important in economics and mathematical optimization. [1]
f and g are lower semi-continuous and ( ) where is the algebraic interior and , where h is some function, is the set {: < +}, or A dom f ∩ cont g ≠ ∅ {\displaystyle A\operatorname {dom} f\cap \operatorname {cont} g\neq \emptyset } where cont {\displaystyle \operatorname {cont} } are the points where the function is ...
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
If the signs differ, then the sequence is concave. In this example, the polygon is negatively oriented, but the determinant for the points F-G-H is positive, and so the sequence F-G-H is concave. The following table illustrates rules for determining whether a sequence of points is convex, concave, or flat: