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In general context, a function may have multiple minimum and maximum values in the domain of function. These minimum and maximum values are “local” minimum and maximum, which belongs to finite sub-intervals within the domain of function. The least minimum and greatest maximum in the domain of function are “global” minimum and maximum respectively in the entire domain of the function. Clearly, least and greatest values are one of the local minimum and maximum values. The minimum and maximum, which are not global, are also known as “relative” minimum and maximum.

Note : This module contains certain concepts relating to continuity, limits and differentiation, which we have not covered in this course. The topic is dealt here because minimum, maximum, least, greatest and range are important attributes of a function and its study is required to complete the discussion on function.

Important observations and definitions

Let us consider a very general graphic representation of a function. Following observations can easily be made by observing the graph :

Graph of a function

There are multiple minimum and maximum values.

1: A function may have local minimum (C, E, G, I) and maximum (B,D,F,H) at more than one point.

2: It is not possible to determine global minimum and maximum unless we know function values corresponding to all values of x in the domain of function. Note that graph above can be defined to any value beyond A.

3: Local minimum at a point (E) can be greater than local maximum at other points (B and H).

4: If function is continuous in an interval, then pair of minimum and maximum in any order occur alternatively (B,C), (C,D), (D,E), (E,F) , (F,G) , (G,H) , (H,I).

5: A function can not have minimum and maximum at points where function is not defined. Consider a rational function, which is not defined at x=1.

f x = 1 x 1 ; x 1

Similarly, a function below is not defined at x=0.

| x=1; x>0 f(x) = ||x = -1; x<0

Graph of function

Function is not at x=0.

Minimum and maximum of function can not occur at points where function is not defined, because there is no function value corresponding to undefined points. We should understand that undefined points or intervals are not part of domain - thus not part of function definition. On the other hand, minimum and maximum are consideration within the domain of function and as such undefined points or intervals should not be considered in the first place. Non-occurance of minimum and maximum in this context, however, has been included here to emphasize this fact.

6: A function can have minimum and maximum at points where it is discontinuous. Consider fraction part function in the finite domain. The function is not continuous at x=1, but minimum occurs at this point (recall its graph).

7: A function can have minimum and maximum at points where it is continuous but not differentiable. In other words, maximum and minimum can occur at corners. For example, modulus function |x| has its only minimum at corner point at x=0 (recall its graph).

Extreme value or extremum

Extreme value or extremum is either a minimum or maximum value. A function, f(x), has a extremum at x=e, if it has either a minimum or maximum value at that point.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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