In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions.

A function (in black) is convex if and only if the region above its graph (in green) is a convex set. A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex. In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points.

2024年9月22日 · Convex functions curve upwards, making them useful for solving problems where you want to find the smallest possible value (like minimizing costs). Concave functions curve downwards, and are often used to find the largest possible value (like maximizing profit).

An easy way to test for both is to connect two points on the curve with a straight line. If the line is above the curve, the graph is convex. If the line is below the curve, the graph is concave.

A function \(f: I \rightarrow \mathbb{R}\) is convex if and only if for every \(\lambda_{i} \geq 0, i=1, \ldots, n\), with \(\sum_{i=1}^{n} \lambda_{i}=1\) \((n \geq 2)\) and for every \(x_{i} \in I\), \(\i=1, \dots, n\), \[f\left(\sum_{i=1}^{n} \lambda_{i} x_{i}\right) \leq \sum_{i=1}^{n} \lambda_{i} f\left(x_{i}\right) .\]

Concave upward is when the slope increases: Concave downward is when the slope decreases: What about when the slope stays the same (straight line)? It could be both! See footnote. Here are some more examples: Concave Upward is also called Convex, or sometimes Convex Downward. Concave Downward is also called Concave, or sometimes Convex Upward.

In this explainer, we will learn how to determine the convexity of a function as well as its inflection points using its second derivative. Before you start with this explainer, you should be confident finding the first and second derivatives of functions using the standard rules for differentiation.

ne functions are the only functions that are both convex and concave. Some quadratic functions: f(x) = xT Qx + cT x + d. { Convex if and only if Q 0. { Strictly convex if and only if Q 0. 0; strictly concave if and only if Q 0. { The proofs are easy if we use the second order characterization of convexity (com-ing up).

2025年1月31日 · A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

Convex curves, or convex functions are curves that curve upwards. Graphically, if a curve is above the line segment connecting any two points on it, the curve is said to be convex. Conversely, concave functions curve downwards. A function is classified as concave if it lies beneath the line segment connecting any two points on it. Determining ...