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- We say that f is concave if f is convex.
www.princeton.edu/~aaa/Public/Teaching/ORF523/S16/ORF523_S16_Lec7_gh.pdf
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Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
- You have two functions f and g, where f''>0 and g''>0. If you take the second derivative of f+g, you get f''+g'', which is positive. So their sum i...
- In order for 𝑓(𝑥) to be concave up, in some interval, 𝑓 ''(𝑥) has to be greater than or equal to 0 (i.e. non-negative) for all 𝑥 in that inter...
- In general, no. Points where f"(x)= 0 are specifically called inflection points. But in the example provided, -2 is also a critical point (Observe...
- Product rule: d/dx[f(x)*g(x)] = f'(x)g(x) + f(x)g'(x) f(x) = (x-1)³ and g(x) = x
Dec 21, 2020 · A function is concave down if its graph lies below its tangent lines. If knowing where a graph is concave up/down is important, it makes sense that the places where the graph changes from one to the other is also important.
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) + +.
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.
Sep 12, 2016 · Your initial statement is wrong: the Hessian does not have to exist for the function to be concave or convex. For example, $|x|$ is convex but does not have derivatives at $0$. A necessary and sufficient condition for a convex/concave function to be strictly convex/concave is that its graph does not contain any line segment.
Sep 5, 2015 · Definition (Concavity/Convexity of a function). Let f: Rn → R. We say that f is concave if for all x, y ∈ Rn and for all λ ∈ [0, 1] we have f(λx + (1 − λ)y) ≥ λf(x) + (1 − λ)f(y). And a function is convex if − f is concave, or f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y). Definition (Quasi-concave/Quasi-convex). Let f: Rn → R.
Calculus conditions for concave functions. (of a single variable). Recall that a real-valued function f is concave if and only if its domain is a convex set A. <n and for all x1 and x2 in A and for all. (1 x1 ( f )x2) (x1) + (1. )f (x2) If A. <, this implies that for all x1 and x2 in A and for all. f (x2) f (x1) + (x2 x1)f 0(x1):
