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In mathematical analysis, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. Image File history File links Please see the file description page for further information. ... A convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on interval I.e. ... Jump to: navigation, search Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... Jump to: navigation, search Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ...

Concave functions

In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. Similarly, a differentiable function f is concave on an interval if its derivative function f ′ is decreasing on that interval: a concave function has a decreasing slope. Jump to: navigation, search Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... Jump to: navigation, search In mathematics, the graph of a function f(x1, x2, ..., xn) is the collection of all tuples (x1, x2, ..., xn, f(x1, ..., xn)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Jump to: navigation, search In mathematics, the derivative is one of the two central concepts of calculus. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the steepness of said line. ... Jump to: navigation, search In mathematics, the graph of a function f(x1, x2, ..., xn) is the collection of all tuples (x1, x2, ..., xn, f(x1, ..., xn)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Jump to: navigation, search In mathematics, the derivative is one of the two central concepts of calculus. ...

A function that is convex is often synonymously called concave upwards, and a function that is concave is often synonymously called concave downward.

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 (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward). Points where concavity changes are inflection points. Jump to: navigation, search Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the gradient of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Stationary points (red pluses) and inflection points (green circles). ...

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. A spatial point is an entity with a location in space but no extent (volume, area or length). ... In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number). ... In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number). ...

Contrary to the impression one may get from a calculus course, differentiability is not essential to these concepts; see convex. In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...

In mathematics, a function f(x) is said to be concave on an interval [a, b] if, for all x,y in [a, b], Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Additionally, f(x) is strictly concave if

A continuous function on C is concave if and only if In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

for any x and y in C.

Equivalently, f(x) is concave on [a, b] iff the function −f(x) is convex on every subinterval of [a, b]. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... A convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on interval I.e. ...

If f(x) is twice-differentiable, then f(x) is concave iff 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. In mathematics, the derivative of a function is one of the two central concepts of calculus. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... A negative number is a number that is less than zero, such as −3. ...

A function is called quasiconcave iff there is an x₀ such that for all x < x₀,f(x) is non-decreasing while for all x > x₀ it is non-increasing. x₀ can also be , making the function non-decreasing (non-increasing) for all x. The opposite of quasiconcave is quasiconvex.

Concave polygons

In a concave polygon, some interior angle is greater than 180°. The extension at that vertex of the line segment making up a side passes through the interior of the polygon. In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ... This article is about angles in geometry. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... In mathematics, a line segment is a part of a line that is bounded by two end points. ...

An example of a concave polygon

A concave polygon is often called re-entrant polygon (but in some cases the latter term has a different meaning). A simple polygon, created with GIMP, in the public domain. ... A simple polygon, created with GIMP, in the public domain. ... Jump to: navigation, search A re-entrant, or concave polygon is one in which at least one interior angle is more than 180 degrees (i. ...

See also

convex In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...