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- 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 function is zero at some point, then that point is a local maximum.
en.wikipedia.org/wiki/Concave_function
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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.
Sep 12, 2016 · For example, |x| | x | is convex but does not have derivatives at 0 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. Share. Cite. answered Sep 12, 2016 at 6:33.
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) + +.
- Definition of Convexity of A Function
- Geometric Interpretation of Convexity
- Sufficient Conditions For Convexity/Concavity
- Properties of Convex Functions
- Solved Problems
Consider a function y = f (x), which is assumed to be continuous on the interval [a, b]. The function y = f (x) is called convex downward (or concave upward) if for any two points x1 and x2 in [a, b], the following inequality holds: If this inequality is strict for any x1, x2 ∈ [a, b], such that x1 ≠ x2, then the function f (x) is called strictly c...
The introduced concept of convexity has a simple geometric interpretation. If a function is convex downward (Figure ), the midpoint of each chord lies above the corresponding point of the graph of the function or coincides with this point. Similarly, if a function is convex upward (Figure ), the midpoint of each chord is located below the correspon...
Suppose that the first derivative of a function exists in a closed interval and the second derivative exists in an open interval Then the following sufficient conditions for convexity/concavity are valid: 1. If for all then the function is convex downward (or concave upward) on the interval 2. If for all then the function is convex upward (or conca...
We list some properties of convex functions assuming that all functions are defined and continuous on the interval 1. If the functions and are convex downward (upward), then any linear combination where , are positive real numbers is also convex downward (upward). 2. If the function is convex downward, and the function is convex downward and non-de...
Solution. Consider an arbitrary combination of these values, such as the following: The graph of such a function is located in the upper half-plane, and the function is strictly decreasing (since ). Given that the function is convex downward. Its schematic view is shown in Figure in the first column and second row. It is clear that the total number...
In this lecture, we shift our focus to the other important player in convex optimization, namely, convex functions. Here are some of the topics that we will touch upon: Convex, concave, strictly convex, and strongly convex functions. First and second order characterizations of convex functions.
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):
