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| | Jerrie Swoboda is what the individual can call me along with I totally dig whom name. What me and my family absolutely love is acting but All of us can't make it all my profession really. The job I've been occupying for years is a real people manager. Guam is even I've always been income. You can sometimes find my website here: http://prometeu.net<br><br>my website; [http://prometeu.net clash of clans hack ifunbox] |
| In [[mathematics]], a '''concave function''' is the [[additive inverse|negative]] of a [[convex function]]. A concave function is also [[synonym]]ously called '''concave downwards''', '''concave down''', '''convex upwards''', '''convex cap''' or '''upper convex'''.
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| ==Definition==
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| A real-valued [[function (mathematics)|function]] ''f'' on an [[interval (mathematics)|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],
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| :<math>f(tx+(1-t)y)\geq t f(x)+(1-t)f(y).</math>
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| A function is called ''strictly concave'' if
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| :<math>f(tx + (1-t)y) > t f(x) + (1-t)f(y)\,</math>
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| for any ''t'' in (0,1) and ''x'' ≠ ''y''.
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| 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'') ).
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| [[Image:ConcaveDef.png]]
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| A function ''f(x)'' is [[quasiconvex function|quasiconcave]] if the upper contour sets of the function <math>S(a)=\{x: f(x)\geq a\}</math> are convex sets.{{sfn|Varian|1992|p=496}}
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| ==Properties==
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| 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.
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| A [[differentiable]] [[graph of a function|function]] ''f'' is concave on an [[interval (mathematics)|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.)
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| 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. [[Point (geometry)|Points]] where concavity changes are [[inflection point]]s.
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| If a convex (i.e., concave upward) function has a "bottom", any [[point (geometry)|point]] at the bottom is a [[Maxima and minima|minimal extremum]]. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a [[Maxima and minima|maximal extremum]].
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| 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 numbers|negative]] then it is strictly concave, but the opposite is not true, as shown by ''f''(''x'') = -''x''<sup>4</sup>.
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| If ''f'' is concave and differentiable, then it is bounded above by its first-order [[Taylor approximation]]:
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| :<math>f(y) \leq f(x) + f'(x)[y-x]</math>{{sfn|Varian|1992|p=489}}
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| A [[continuous function]] on ''C'' is concave [[if and only if]] for any ''x'' and ''y'' in ''C''
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| :<math>f\left( \frac{x+y}2 \right) \ge \frac{f(x) + f(y)}2</math>
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| If a function ''f'' is concave, and ''f''(0) ≥ 0, then ''f'' is [[subadditivity|subadditive]]. Proof:
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| * since ''f'' is concave, let ''y'' = 0, <math>f(tx) = f(tx+(1-t)\cdot 0) \ge t f(x)+(1-t)f(0) \ge t f(x)</math>
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| * <math>f(a) + f(b) = f \left((a+b) \frac{a}{a+b} \right) + f \left((a+b) \frac{b}{a+b} \right)
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| \ge \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) = f(a+b)</math>
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| ==Examples==
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| * The functions <math>f(x)=-x^2</math> and <math>g(x)=\sqrt{x}</math> are concave on their domains, as their second derivatives <math>f''(x) = -2</math> and <math>g''(x) = -\frac{1}{4 x^{1.5}}</math> are always negative.
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| * Any [[affine function]] <math>f(x)=ax+b</math> is both (non-strictly) concave and convex.
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| * The [[sine]] function is concave on the interval <math>[0, \pi]</math>.
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| * The function <math>f(x) = \log |B|</math>, where <math>|B|</math> is the [[determinant]] of a [[nonnegative-definite matrix]] ''B'', is concave.<ref name="Cover 1988">{{cite journal|author=[[Thomas M. Cover]] and J. A. Thomas| title=Determinant inequalities via information theory| journal=[[SIAM Journal on Matrix Analysis and Applications]]| year=1988| volume=9|number=3| pages=384–392}}</ref>
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| * Practical example: rays bending in [[Computation of radiowave attenuation in the atmosphere]].
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| ==See also==
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| * [[Concave polygon]]
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| * [[Convex function]]
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| * [[Jensen's inequality]]
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| * [[Logarithmically concave function]]
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| * [[Quasiconcave function]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book|last=Crouzeix|first=J.-P.|chapter=Quasi-concavity|title=The New Palgrave Dictionary of Economics|editor-first=Steven N.|editor-last=Durlauf|editor2-first=Lawrence E<!-- . -->|editor2-last=Blume|publisher=Palgrave Macmillan|year=2008|edition=Second|pages=|url=http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008|doi=10.1057/9780230226203.1375|ref=harv}}
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| *{{cite book |title=Engineering Optimization: Theory and Practice|first=Singiresu S.|last=Rao|
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| publisher=John Wiley and Sons|year=2009|isbn=0-470-18352-7|page=779}}
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| *{{cite book |ref=harv |last=Varian |first=Hal R. |authorlink=Hal Varian |title=Microeconomic Analysis |year=1992 |edition=Third |publisher=W.W. Norton and Company}}
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| {{DEFAULTSORT:Concave Function}}
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| [[Category:Types of functions]]
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| [[Category:Convex analysis]]
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| [[de:Konvexe und konkave Funktionen]]
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| [[ja:凹関数]]
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Jerrie Swoboda is what the individual can call me along with I totally dig whom name. What me and my family absolutely love is acting but All of us can't make it all my profession really. The job I've been occupying for years is a real people manager. Guam is even I've always been income. You can sometimes find my website here: http://prometeu.net
my website; clash of clans hack ifunbox