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- A function g (x) is called concave on an interval if, for any two points x_1 x1 and x_2 x2 in the interval and any lambda in [0, 1] λ∈ [0,1], the following holds: g (lambda x_1 + (1-lambda) x_2) geq lambda g (x_1) + (1-lambda) g (x_2) g(λx1 +(1−λ)x2) ≥ λg(x1)+(1−λ)g(x2)
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In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex.
Let f : S ⊂Rn →R be a concave (convex) function, and let g : R →R be concave (convex) and increasing. Then (f g) : S ⊂R n →R is a concave (convex) function.
the convexity of particular sets. Given the graph of a function, the hypograph of f, written hypf, is the set of points that lies on or below the graph of f, while the epigraph of f, written epif, is the set of points that lies on or above the graph of f.2 Formally, epif= f(x;y) 2RN+1: y f(x)g; hypf= f(x;y) 2RN+1: y f(x)g: Theorem 4.
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Dec 21, 2020 · The graph of a function \(f\) is concave up when \(f'\) is increasing. That means as one looks at a concave up graph from left to right, the slopes of the tangent lines will be increasing. Consider Figure \(\PageIndex{1}\), where a concave up graph is shown along with some tangent lines.
A function f is concave if the 2nd derivative f’’ is negative (f’’ < 0). Graphically, a concave function opens downward, and water poured onto the curve would roll off. A function f is convex if f’’ is positive (f’’ > 0). A convex function opens upward, and water poured onto the curve would fill it.
Oct 17, 2016 · Suppose that $f: \mathbb{R}\to \mathbb{R}$ is convex function and $g: \mathbb{R}\to \mathbb{R}$ is concave function. What can we say about their composition $g\circ f$ and $f\circ g$? Are they conv...
If f' (x) is decreasing over an interval, then the graph of f (x) is concave down over the interval. Given a graph of f (x) or f' (x), as well as the facts above, it is relatively simple to determine the concavity of a function.
