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es upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definition 2 search 3 examples 4 functions of more than one variable 5 maxima or minima of a functional 6 in relation to sets 7 argument of the maximum 8 see also 9 notes 10 references 11 external links toggle the table of contents maximum and minimum 55 languages العربية azərbaycanca български català čeština чӑвашла cymraeg dansk deutsch ελληνικά esperanto español eesti euskara فارسی suomi français galego עברית हिन्दी magyar հայերեն bahasa indonesia íslenska italiano 日本語 қазақша ಕನ್ನಡ 한국어 lombard latviešu nederlands norsk nynorsk norsk bokmål polski português română русский simple english slovenčina slovenščina chishona shqip svenska தமிழ் ไทย tagalog türkçe татарча tatarça українська oʻzbekcha ўзбекча tiếng việt 閩南語 bân lâm gí 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikibooks wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from maxima and minima largest and smallest value taken by a function at a given point extreme value redirects here for other uses see extreme value disambiguation maximum and minimum redirect here for other uses see maximum disambiguation and minimum disambiguation local and global maxima and minima for cos 3 π x x 0 1 x 1 1 global and local extremums of a function in mathematical analysis the maximum and minimum a of a function are respectively the greatest and least value taken by the function known generically as extrema b they may be defined either within a given range the local or relative extrema or on the entire domain the global or absolute extrema of a function 1 2 3 pierre de fermat was one of the first mathematicians to propose a general technique adequality for finding the maxima and minima of functions as defined in set theory the maximum and minimum of a set are the greatest and least elements in the set respectively unbounded infinite sets such as the set of real numbers have no minimum or maximum in statistics the corresponding concept is the sample maximum and minimum definition edit a real valued function f defined on a domain x has a global or absolute maximum point at x if f x f x for all x in x similarly the function has a global or absolute minimum point at x if f x f x for all x in x the value of the function at a maximum point is called the maximum value of the function denoted max f x displaystyle max f x and the value of the function at a minimum point is called the minimum value of the function denoted min f x displaystyle min f x for clarity symbolically this can be written as follows x 0 x displaystyle x_ 0 in x is a global maximum point of function f x r displaystyle f x to mathbb r if x x f x 0 f x displaystyle forall x in x f x_ 0 geq f x the definition of global minimum point also proceeds similarly if the domain x is a metric space then f is said to have a local or relative maximum point at the point x if there exists some ε 0 such that f x f x for all x in x within distance ε of x similarly the function has a local minimum point at x if f x f x for all x in x within distance ε of x a similar definition can be used when x is a topological space since the definition just given can be rephrased in terms of neighbourhoods mathematically the given definition is written as follows let x d x displaystyle x d_ x be a metric space and function f x r displaystyle f x to mathbb r then x 0 x displaystyle x_ 0 in x is a local maximum point of function f displaystyle f if ε 0 displaystyle exists varepsilon 0 such that x x d x x x 0 ε f x 0 f x displaystyle forall x in x d_ x x x_ 0 varepsilon implies f x_ 0 geq f x the definition of local minimum point can also proceed similarly in both the global and local cases the concept of a strict extremum can be defined for example x is a strict global maximum point if for all x in x with x x we have f x f x and x is a strict local maximum point if there exists some ε 0 such that for all x in x within distance ε of x with x x we have f x f x note that a point is a strict global maximum point if and only if it is the unique global maximum point and similarly for minimum points a continuous real valued function with a compact domain always has a maximum point and a minimum point an important example is a function whose domain is a closed and bounded interval of real numbers see the graph above search edit finding global maxima and minima is the goal of mathematical optimization if a function is continuous on a closed interval then by the extreme value theorem global maxima and minima exist furthermore a global maximum or minimum either must be a local maximum or minimum in the interior of the domain or must lie on the boundary of the domain so a method of finding a global maximum or minimum is to look at all the local maxima or minima in the interior and also look at the maxima or minima of the points on the boundary and take the greatest or least one for differentiable functions fermat s theorem states that local extrema in the interior of a domain must occur at critical points or points where the derivative equals zero 4 however not all critical points are extrema one can often distinguish whether a critical point is a local maximum a local minimum or neither by using the first derivative test second derivative test or higher order derivative test given sufficient differentiability 5 for any function that is defined piecewise one finds a maximum or minimum by finding the maximum or minimum of each piece separately and then seeing which one is greatest or least examples edit the global maximum of x x occurs at x e function maxima and minima x 2 unique global minimum at x 0 x 3 no global minima or maxima although the first derivative 3 x 2 is 0 at x 0 this is an inflection point 2nd derivative is 0 at that point x x displaystyle sqrt x x unique global maximum at x e see figure at right also see steiner s calculus problem x x unique global maximum over the positive real numbers at x 1 e x 3 3 x first derivative x 2 1 and second derivative 2 x setting the first derivative to 0 and solving for x gives stationary points at 1 and 1 from the sign of the second derivative we can see that 1 is a local maximum and 1 is a local minimum this function has no global maximum or minimum x global minimum at x 0 that cannot be found by taking derivatives because the derivative does not exist at x 0 cos x infinitely many global maxima at 0 2 π 4 π and infinitely many global minima at π 3 π 5 π 2 cos x x infinitely many local maxima and minima but no global maximum or minimum cos 3 π x x with 0 1 x 1 1 global maximum at x 0 1 a boundary a global minimum near x 0 3 a local maximum near x 0 6 and a local minimum near x 1 0 see figure at top of page x 3 3 x 2 2 x 1 defined over the closed interval segment 4 2 local maximum at x 1 15 3 local minimum at x 1 15 3 global maximum at x 2 and global minimum at x 4 for a practical example 6 assume a situation where someone has 200 displaystyle 200 feet of fencing and is trying to maximize the square footage of a rectangular enclosure where x displaystyle x is the length y displaystyle y is the width and x y displaystyle xy is the area 2 x 2 y 200 displaystyle 2x 2y 200 2 y 200 2 x displaystyle 2y 200 2x 2 y 2 200 2 x 2 displaystyle frac 2y 2 frac 200 2x 2 y 100 x displaystyle y 100 x x y x 100 x displaystyle xy x 100 x the derivative with respect to x displaystyle x is d d x x y d d x x 100 x d d x 100 x x 2 100 2 x displaystyle begin aligned frac d dx xy frac d dx x 100 x frac d dx left 100x x 2 right 100 2x end aligned setting this equal to 0 displaystyle 0 0 100 2 x displaystyle 0 100 2x 2 x 100 displaystyle 2x 100 x 50 displaystyle x 50 reveals that x 50 displaystyle x 50 is our only critical point now retrieve the endpoints by determining the interval to which x displaystyle x is restricted since width is positive then x 0 displaystyle x 0 and since x 100 y displaystyle x 100 y that implies that x 100 displaystyle x 100 plug in critical point 50 displaystyle 50 as well as endpoints 0 displaystyle 0 and 100 displaystyle 100 into x y x 100 x displaystyle xy x 100 x and the results are 2500 0 displaystyle 2500 0 and 0 displaystyle 0 respectively therefore the greatest area attainable with a rectangle of 200 displaystyle 200 feet of fencing is 50 50 2500 displaystyle 50 times 50 2500 6 functions of more than one variable edit main article second partial derivative test peano surface a counterexample to some criteria of local maxima of the 19th century the global maximum is the point at the top counterexample the red dot shows a local minimum that is not a global minimum for functions of more than one variable similar conditions apply for example in the enlargeable figure on the right the necessary conditions for a local maximum are similar to those of a function with only one variable the first partial derivatives as to z the variable to be maximized are zero at the maximum the glowing dot on top in the figure the second partial derivatives are negative these are only necessary not sufficient conditions for a local maximum because of the possibility of a saddle point for use of these conditions to solve for a maximum the function z must also be differentiable throughout the second partial derivative test can help classify the point as a relative maximum or relative minimum in contrast there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema for example if a bounded differentiable function f defined on a closed interval in the real line has a single critical point which is a local minimum then it is also a global minimum use the intermediate value theorem and rolle s theorem to prove this by contradiction in two and more dimensions this argument fails this is illustrated by the function f x y x 2 y 2 1 x 3 x y r displaystyle f x y x 2 y 2 1 x 3 qquad x y in mathbb r whose only critical point is at 0 0 which is a local minimum with f 0 0 0 however it cannot be a global one because f 2 3 5 maxima or minima of a functional edit if the domain of a function for which an extremum is to be found consists itself of functions i e if an extremum is to be found of a functional then the extremum is found using the calculus of variations in relation to sets edit maxima and minima can also be defined for sets in general if an ordered set s has a greatest element m then m is a maximal element of the set also denoted as max s displaystyle max s furthermore if s is a subset of an ordered set t and m is the greatest element of s with respect to order induced by t then m is a least upper bound of s in t similar results hold for least element minimal element and greatest lower bound the maximum and minimum function for sets are used in databases and can be computed rapidly since the maximum or minimum of a set can be computed from the maxima of a partition formally they are self decomposable aggregation functions in the case of a general partial order the least element i e one that is less than all others should not be confused with a minimal element nothing is lesser likewise the greatest element of a partially ordered set poset is an upper bound of the set which is contained within the set whereas a maximal element m of a poset a is an element of a such that if m b for any b in a then m b any least element or greatest element of a poset is unique but a poset can have several minimal or maximal elements if a poset has more than one maximal element then these elements will not be mutually comparable in a totally ordered set or chain all elements are mutually comparable so such a set can have at most one minimal element and at most one maximal element then due to mutual comparability the minimal element will also be the least element and the maximal element will also be the greatest element thus in a totally ordered set we can simply use the terms minimum and maximum if a chain is finite then it will always have a maximum and a minimum if a chain is infinite then it need not have a maximum or a minimum for example the set of natural numbers has no maximum though it has a minimum if an infinite chain s is bounded then the closure cl s of the set occasionally has a minimum and a maximum in which case they are called the greatest lower bound and the least upper bound of the set s respectively argument of the maximum edit this section is an excerpt from arg max edit as an example both unnormalised and normalised sinc functions above have argmax displaystyle operatorname argmax of 0 because both attain their global maximum value of 1 at x 0 the unnormalised sinc function red has arg min of 4 49 4 49 approximately because it has 2 global minimum values of approximately 0 217 at x 4 49 however the normalised sinc function blue has arg min of 1 43 1 43 approximately because their global minima occur at x 1 43 even though the minimum value is the same 7 in mathematics the arguments of the maxima abbreviated arg max or argmax and arguments of the minima abbreviated arg min or argmin are the input points at which a function output value is maximized and minimized respectively 8 while the arguments are defined over the domain of a function the output is part of its codomain see also edit derivative test infimum and supremum limit superior and limit inferior maximum minimums identity mechanical equilibrium mex mathematics saddle point sample maximum and minimum notes edit pl maxima and minima or maximums and minimums sg extremum references edit stewart james 2008 calculus early transcendentals 6th ed brooks cole isbn 978 0 495 01166 8 larson ron edwards bruce h 2009 calculus 9th ed brooks cole isbn 978 0 547 16702 2 thomas george b weir maurice d hass joel 2010 thomas calculus early transcendentals 12th ed addison wesley isbn 978 0 321 58876 0 weisstein eric w minimum mathworld wolfram com retrieved 2020 08 30 weisstein eric w maximum mathworld wolfram com retrieved 2020 08 30 1 2 garrett paul minimization and maximization refresher the unnormalized sinc function archived 2017 02 15 at the wayback machine university of sydney for clarity we refer to the input x as points and the output y as values compare critical point and critical value external links edit wikimedia commons has media related to extrema calculus look up maxima minima or extremum in wiktionary the free dictionary thomas simpson s work on maxima and minima at convergence application of maxima and minima with sub pages of solved problems jolliffe arthur ernest 1911 maxima and minim...