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- Concavity refers to the curvature of a function and indicates whether the function is bending upwards or downwards. Understanding concavity helps identify the behavior of a function's graph, especially when analyzing maximum and minimum points through symbolic differentiation.
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Illustrated definition of Concave: Curved inwards. Example: A polygon (which has straight sides) is concave when there are dents or indentations...
- Convex
Curved outwards. Example: A polygon (which has straight...
- Convex
Concave. A shape is concave if part of it is pushed inwards, or "caved in". It is the opposite of convex, where all parts of the figure point outwards. In a geometric figure, it is concave if you can draw at least one line segment between two points on it that is outside the figure.
The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below. Concave up.
Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
Theorem 3. Let C R be an open interval. 1. f: C!R is concave i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a: For strict concavity, the inequalities are strict. 2. f: C!R is convex i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a:
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Nov 14, 2023 · A concave function is a function for which any secant line (a line connecting two points on the function) lies below the function. The shape of a concave function looks approximately like...
5 days ago · A set in is concave if it does not contain all the line segments connecting any pair of its points. If the set does contain all the line segments, it is called convex.
