5.6 Minimum and maximum values

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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 :

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.

$f\left(x\right)=\frac{1}{x-1};\phantom{\rule{1em}{0ex}}x\ne 1$

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

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

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x