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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.
A function is concave (convex) if the graph of the function is always above (below) any chord (line segment between two points in the graph). Remark 4. f concave ⇔−f convex.
Aug 24, 2022 · My question is, if we take a concave function like the picture below, and take the negative of it, will it become convex? Since the appearance will be U-shaped just like a convex function. I understand that the inequality for concave function still holds if we negate the function.
- Convex Sets
- Concave and Convex Functions
- Twice-differentiable Concave and Convex Functions
For n = 1, the definition coincides with the definition of an interval: a set of numbers is convex if and only if it is an interval. For n = 2, two examples are given in the following figures. The set in the first figure is convex, because every line segment joining a pair of points in the set lies entirely in the set. The set in the second figure ...
More precisely, we can make the following definition (which is again essentially the same as the corresponding definition for a function of a single variable). Note that only functions defined on convex sets are covered by the definition.
To determine whether a twice-differentiable function of many variables is concave or convex, we need to examine all its second partial derivatives. We call the matrix of all the second partial derivatives the Hessianof the function. We can determine the concavity/convexity of a function by determining whether the Hessian is negative or positive sem...
f(b)(c a) f(a)(c b) + f(c)(b a); which (since c a > 0) holds i. f(b)baf(a) +aaf(c):Take = (c b)=(c a) 2 (0; 1) and verify. that, indeed, b = a + (1 )c. Then the last inequal. ty holds since f is concave. Conversely, the preceding argument shows that if the rst inequality in (1) holds then f is.
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A function ’is concave if every chord lies below the graph of ’. Another fundamental geometric property of convex functions is that each tangent line lies entirely below the graph of the function.
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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.
