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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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Concavity refers to the direction of the curvature of a function's graph. It indicates whether the graph is bending upwards (concave up) or downwards (concave down) and is closely related to the second derivative of the function.
Sep 9, 2023 · Definition: An object or a function is concave if it curves inward. In simple terms, it’s hollow or bowed in, much like a cave. Everyday Examples: A bowl. A satellite dish. A spoon’s interior. Skateboard ramps. A pie with a slice taken out of it. Convex.
A function $f(x)$ is said to be quasi-concave if its domain and all its $\alpha$-superlevel sets defined as $$\mathcal{S}_\alpha \triangleq\{x|x\in dom f, f(x)\geq\alpha\}$$ are convex for every $\alpha$.
Then, if f′′(c) > 0 f ″ ( c) > 0, the graph of f f is concave upward at (c, f(c)) ( c, f ( c)); if f′′(c) < 0 f ″ ( c) < 0, the graph of f f is concave downward at (c, f(c)) ( c, f ( c)). To convince the curious among you of this, let us prove the first part of the theorem above:
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.
