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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) + +.
It is also possible to characterize concavity or convexity of functions in terms of the convexity of particular sets. Given the graph of a function, the hypograph of f,
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May 22, 2024 · Solved Examples on Concavity. Example 1: Determine the intervals where the function f(x)=x 3 −6x 2 +9x+15 is concave up and concave down. First Derivative: f'(x)=3x 2-12x + 9. Second Derivative: f"(x)=6x-12. Find Critical Points for Concavity: Set the second derivative equal to zero to find potential points of inflection: 6x-12=0. x=12/6=2
Sep 22, 2024 · Examples of Concave Functions. Some examples of concave functions are: g(x) = - x 2; g(x) = log x; g(x) = -e x; Graphical Representation of Concave Functions. Mathematically, a function f(x) is concave if for any two points x 1 and x 2 and any λ ∈ [0, 1], the following inequality holds: f(λx 1 + (1 − λ)x 2) ≥ λf(x 1) + (1 − λ)f(x 2 ...
Jul 30, 2019 · Here is a github repo on finding the concave hull for a set of points using python. My recommendation to you is the following. Create a set of points using the endpoints of each line. Then use the linked to code to generate a concave hull for these points, with some guess for the value of alpha.
Anatomy of a Function def main(): mid = average(10.6, 7.2) print(mid) def average(a, b): sum = a + b return sum / 2 Think/Pair/Share: Find the function definition, function name, parameter(s), and return value in average.
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Advanced Functions. In this chapter, we go beyond the basics of using functions. I’ll assume you can define and work with functions taking default arguments: > def foo(a, b, x=3, y=2): ... return (a+b)/(x+y) ... > foo(5, 0) 1.0. > foo (10, 2, y=3) 2.0.
