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Topic: Concave function

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	PlanetMath: convex function
		is a strictly convex function, or a strictly concave function, respectively.
		We may generalize the above definition of a convex function to an that of an extended real-valued function whose domain is not necessarily a convex set.
		This is version 23 of convex function, born on 2001-10-15, modified 2007-08-01.
	planetmath.org /encyclopedia/ConvexFunction.html (303 words)

	PlanetMath: concavity of sine function
		Theorem 1 The sine function is concave on the interval
		"concavity of sine function" is owned by rspuzio.
		This is version 5 of concavity of sine function, born on 2007-04-28, modified 2007-08-01.
	planetmath.org /encyclopedia/ConcavityOfSineFunction.html (106 words)

	Concave - Wikipedia, the free encyclopedia
		Concave lens, a lens with inward-curving (concave) surfaces.
		Concave function, a type of function which is related to convex functions.
		In addition, the term concave upwards is used for convex functions, and concave downwards for concave functions.
	en.wikipedia.org /wiki/Concave (126 words)

	Course diary for Math 135, section F2, summer 2006
		The function increases in the intervals [-2,0] and [1,infinity) and decreases in the intervals (-infinity,-1] and [0,1].
		The function is increasing in the interval (-infinity,0] and decreasing in the interval [0,+infinity).
		The function is decreasing in the interval [-1,1].
	math.rutgers.edu /~greenfie/summer_135/diary2.html (11281 words)

	concave - Search Results - MSN Encarta
		Concave, shape of a surface curving inward, or away from the eye.
		The word concave means curving in or hollowed inward.
		In mathematics, a real-valued function f defined on an interval (or on any convex set C of some vector space) is called concave, if for any two points x and y in its domain C and any t in [0,1], we...
	encarta.msn.com /encnet/refpages/search.aspx?q=concave (208 words)

	Concavity and the Second Derivative Test - HMC Calculus Tutorial
		The graph of a function f is concave upward at the point (c, f(c)) if f'(c) exists and if for all x in some open interval containing c, the point (x, f(x)) on the graph of f lies above the corresponding point on the graph of the tangent line to f at c.
		The graph of a function f is concave downward at the point (c, f(c))$"> if f'(c) exists and if for all x in some open interval containing c, the point (x, f(x)) on the graph of f lies below the corresponding point on the graph of the tangent line to f at c.
		The Second Derivative Test relates the concepts of critical points, extreme values, and concavity to give a very useful tool for determining whether a critical point on the graph of a function is a relative minimum or maximum.
	www.math.hmc.edu /calculus/tutorials/secondderiv (689 words)

	Convex and Concave Functions (Site not responding. Last check: 2007-10-16)
		It is important not to confuse the concepts of convex sets and convex functions, especially since one of the criteria for a function to be convex is that its epigraph form a convex set.
		Convex and concave functions are especially nice for optimization purposes since they have a unique relative min and max respectively and hence this relative min or max is also its absolute min or max.
		Although functions encountered in practice may not be convex or concave, they can sometimes be closely approximated by convex or concave functions.
	www.saintmarys.edu /~psmith/338act20.html (410 words)

	Body
		Suppose the graph of a function f is either concave upward or concave downward on an open interval I.
		Suppose the graph of f is concave downward on (a, b).
		Therefore, if a function f is a twice differentiable function, then at any point, where the second derivative is positive, the graph of f is concave up there, and at any point with negative second derivative there, the graph of f is concave down there.
	www.math.nus.edu.sg /~matngtb/Calculus/Concavity_line/concave.htm (3111 words)

	Economic interpretation of calculus operations - univariate
		Recall from past section on linear functions that the slope of a horizontal line or function is equal to zero.
		Therefore, we would expect the underlying function to be one where the first derivative is zero at the turning point, with a positive second derivative in the neighborhood of the turning point, indicating an increasing slope.
		Even though MC is the function for the slope of total cost, ignore that and treat it as a stand-alone function, and take the first and second order derivatives according to the steps of optimization.
	www.columbia.edu /itc/sipa/math/calc_econ_interp_u.html (2164 words)

	Concave function - Wikipedia, the free encyclopedia
		In other words, a function is concave if and only if its epigraph (the set of points lying on or above the graph) is a concave set.
		A differentiable function f is concave on an interval if its derivative function f ′ is monotone decreasing on that interval: a concave function has a decreasing slope.
		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.
	en.wikipedia.org /wiki/Concave_function (386 words)

	Function Behavior - Background
		Alternate: for any small number e, the slope of the function at c-e is equal to the slope of the function at c+e.
		Since the first derivative of a function is the slope of the function at any point; if the first derivative is positive, then the function is increasing.
		For example, a function from the Newtonian motion equations on an interval of downward concavity implies particle is decelerating.
	www.krellinst.org /AiS/textbook/unit2/example_projects/starter/math/calc/function/function.back.html (1557 words)

	9.2 Functions (Site not responding. Last check: 2007-10-16)
		The argument is an affine function or a variable.
		Variables and affine functions admit single-argument indexing of the four types described in section 2.4.
		Piecewise-linear functions admit single-argument indexing of the four types described in section 2.4.
	www.ee.ucla.edu /~vandenbe/cvxopt/doc/s-functions.html (649 words)

	MATH 150 SUPPLEMENTAL NOTES 13
		A fringe benefit of determining if a function is increasing or decreasing on an interval is the ability to determine if the critical point is a local max or min.
		After determining the intervals where the function is concave up or concave down, we can determine the points of inflection.
		For the following function, determine where it is increasing and decreasing, all extrema, where it is concave up and concave down, and points of inflection.
	faculty.eicc.edu /bwood/math150supnotes/supplemental13.htm (1874 words)

	Concavity
		If the curve of the function f always remains above the tangent lines for every point in the interval I, we say that the curve is concave upward on that interval.
		In such cases, the concavity of the function is determined by its concavity before and after that point.
		When the concavity of the function changes after and before the point, it is a point of inflection.
	library.thinkquest.org /C006002/Pages/Concavity.htm (245 words)

	Sign of 2nd derivative, Maths First, Institute of Fundamental Sciences, Massey University
		Convince yourself that the graph of the given function f is concave up where the derivative f ' (the slope of the tangent) is an increasing function, and concave down where the derivative f ' is a decreasing function.
		Continue until you are convinced that, in all cases, the graph of the given function f is concave up where the derivative f ' (the slope of the tangent) is an increasing function, and concave down where the derivative f ' is a decreasing function.
		Note that the function is shown on the left, the first derivative in the middle and the second derivative on the left.
	mathsfirst.massey.ac.nz /Calculus/Sign2ndDer/Sign2DerPOI.htm (1208 words)

	Untitled
		The graph of the function is concave down for x > = 1 and x < 4/3.
		The function equals zero at x = 1 or x = -1 since the numerator x^2 - 1 can be factored into (x + 1)(x 1).
		Since the function is a polynomial, the function and all its derivatives are defined everywhere.
	www.gomath.com /category/A_Functions/2.html (1442 words)

	Models - Operations Research Models and Methods
		A function of a single variable with a decreasing derivative.
		A concave function in the objective function of a maximization problem can be represented by the sum of several linear expressions with a piecewise linear approximation (3 for the figure).
		A convex function in the objective function of a minimization problem can be represented by the sum of several linear expressions with a piecewise linear approximation (3 for the figure).
	www.me.utexas.edu /~jensen/ORMM/models/unit/integer/subunits/terminology/index.html (363 words)

	Math Forum - Ask Dr. Math
		If it's positive there, the function is concave up (think of an upward wind - the positive direction - blowing on the graph of the function), and if it's negative, it's concave down (a downward wind).
		Since you also want to know all values of x for which this function is concave down, we'll kill two math birds with one stone.
		We've got a function that's concave down eveywhere except one point, and it has a local maximum at (-8, 20) and a local minimum at (0, 0).
	mathforum.org /library/drmath/view/53393.html (538 words)

	Visual Calculus / Graphs and Derivatives
		Some examples of finding graphically where a given function is concave upward or concave downward are given.
		f is concave downward on (c, a) and is concave upward on (a, d).
		Quiz on determining which graph is the graph of a function, its derivative and its 2nd derivatives.
	archives.math.utk.edu /visual.calculus/3/graphing.14/index.html (455 words)

	Models - Operations Research Models and Methods
		The objective and constraint functions are scalar quantities that vary with the decision vector.
		If the objective function is a convex and the feasible region defines a convex set, every local minimum is a global minimum.
		If the objective function is not a convex function, a local minima may or may not be global minimum.
	www.me.utexas.edu /~jensen/ORMM/models/unit/nonlinear/subunits/terminology/index.html (396 words)

	Gender Inequality in Human Development
		We define one monotonic increasing function to be more concave than another if the former can be expressed as a concave monotonic increasing transform of the latter.
		It was also established as an inequality concerning convex functions in Hardy, Littlewood and Pólya (1952: 75-6).
		This is an increasing concave function for all values (positive and negative) of V, because (1-v) < 0.
	hdr.undp.org /docs/publications/ocational_papers/oc19c.htm (697 words)

	All Elementary Mathematics - Study Guide - Principles of analysis - Convexity, concavity and inflexion points of a ...
		Sufficient condition of concavity (convexity) of a function.
		If a function changes a convexity to a concavity or vice versa at passage through some point, then this point is called an inflexion point an inflexion point.
		This function is concave at x > 0 and convex at x < 0.
	www.bymath.com /studyguide/ana/sec/conc_conv.htm (346 words)

	Calculus I Notes, Section 4-3
		The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant.
		In particular it tells us when the function is concave up, concave down, or there is a point of inflection.
		If the second derivative is positive then the rate of change of the first derivative is positive which means that the function is concave up.
	www.blc.edu /faculty/rbuelow/calc/nt4-3.html (503 words)

	Quick Review 4-2
		An inflection point is the point on the curve where the graph changes from concave up to concave down, or vice versa.
		The sign of the second derivative indicates whether a function is concave up or concave down.
		Use the second derivative as a test function for concave up and concave down.
	www.ltu.edu /courses/lowry/techcalc/mod4-2.htm (739 words)

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