Concave function - Wikipedia
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.
3.4: Concavity and the Second Derivative - Mathematics LibreTexts
2020年12月21日 · 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.
Concavity - Desmos
This graph determines the concavity and inflection points for any function equal to f(x). Green = concave up, red = concave down, blue bar = inflection point.
Concave Function - GeeksforGeeks
2024年7月30日 · In a single variable context, a function f(x) is concave if its graph lies above or on any line segment connecting two points on the graph. More formally, for any two points x₁ and x₂ in the domain and any t ∈ [0,1], a concave function satisfies f(tx₁ + (1-t)x₂) ≥ tf(x₁) + (1-t)f(x₂).
Concave Upward and Downward - Math is Fun
When the slope continually increases, the function is concave upward. When the slope continually decreases , the function is concave downward . Taking the second derivative actually tells us if the slope continually increases or decreases.
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, 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,
Concavity - Math.net
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.
Concave Function -- from Wolfram MathWorld
2024年10月21日 · Concave Function. A function is said to be concave on an interval if, for any points and in , the function is convex on that interval (Gradshteyn and Ryzhik 2000).
5.4: Concavity and Inflection Points - Mathematics LibreTexts
2020年12月21日 · If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
Section 4.6 : The Shape of a Graph, Part II - Pauls Online Math Notes
2022年11月16日 · In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down.