http___en.wikipedia.org_wiki_Concave_function.xml
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Изпъкнала функция Изпъкнала функция е вид нелинейна функция в математиката. Съдържание Определение[редактиране | edit source] Нека в някакъв интервал (x1;x2) имаме дефинирана непрекъсната функция y(x), представена с крива, еднопосочно "огъната" условно само "нагоре". Нека А и Б са стойностите на тази функция съответно в точките x1 и x2. Тогава, ако графиката на функцията y(x) е под линията АБ, функцията се нарича изпъкнала. Строгото определение за вдлъбната и изпъкнала функция се формира така: Изпъкнала функция[редактиране | edit source] Функцията y(x) се нарича изпъкнала в даден интервал, ако за всеки две точки x1 и x2 от него е изпълнено неравенството: Прието е линейната функция да бъде едновременно изпъкнала и вдлъбната. Поради това знакът за равенство присъства в горните дефиниции. Свойства[редактиране | edit source] - Ако функцията y(x) притежава втора производна, която е положителна (f′′(x)>0) в дадения интервал, то функцията е изпъкнала в този интервал. - Точките, отделящи изпъкнала от вдлъбната част на графиката на функция, се наричат инфлексни точки за графиката на функцията.
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Concave function In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex. Definition[edit] A real-valued function f on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any x and y in the interval and for any t in [0,1], A function is called strictly concave if for any t in (0,1) and x ≠ y. For a function f:R→R, this definition merely states that for every z between x and y, the point (z, f(z) ) on the graph of f is above the straight line joining the points (x, f(x) ) and (y, f(y) ). A function f(x) is quasiconcave if the upper contour sets of the function are convex sets.[1] Properties[edit] A function f(x) is concave over a convex set if and only if the function −f(x) is a convex function over the set. A differentiable function f is concave on an interval if its derivative function f ′ is monotonically decreasing on that interval: a concave function has a decreasing slope. ("Decreasing" here means non-increasing, rather than strictly decreasing, and thus allows zero slopes.) For a twice-differentiable function f, if the second derivative, f ′′(x), is positive (or, if the acceleration is positive), then the graph is convex; if f ′′(x) is negative, then the graph is concave. Points where concavity changes are inflection points. If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum. If f(x) is twice-differentiable, then f(x) is concave if and only if f ′′(x) is non-positive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x4. If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation: A continuous function on C is concave if and only if for any x and y in C If a function f is concave, and f(0) ≥ 0, then f is subadditive. Proof: - since f is concave, let y = 0, Examples[edit] - The functions and are concave on their domains, as are their second derivatives and are always negative. - Any affine function is both (non-strictly) concave and convex. - The sine function is concave on the interval . - The function , where is the determinant of a nonnegative-definite matrix B, is concave.[3] - Practical example: rays bending in computation of radiowave attenuation in the atmosphere. See also[edit] - Concave polygon - Convex function - Jensen's inequality - Logarithmically concave function - Quasiconcave function Notes[edit] - ^ Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory". SIAM Journal on Matrix Analysis and Applications 9 (3): 384–392. References[edit] - Crouzeix, J.-P. "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. - Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. - Varian, Hal R. (1992). Microeconomic Analysis (Third ed.). W.W. Norton and Company.