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Least and greatest function values are characterizing aspects of a function. In particular, they allow us to determine range of function if function is continuous. Determination of these values, however, is not straight forward as there may be large numbers of minimum and maximum through out the domain of the function. It is difficult to say which of these are least or greatest of all. Two things simplify our analysis : (i) domain of investigation is finite and (ii) function is monotonic in sub-intervals within domain.

The study of least and greatest function values in this module is targeted to determine range of function. If “A” and “B” be the least and greatest values of a continuous function in a finite interval, then range of the function is given by :

Range = [ least, greatest ] = [ a,b ]

We should note that determining range is a comparatively more difficult proposition than determining domain. Recall that we need to solve given function for x to determine range. This solution, however, is not always explicit. As such, we may be stuck with problem of finding range of more complex functions – particularly those, which involves transcendental functions.

Further, we need to underline one important aspect, while evaluating range of a composite function. Range of a composite function is evaluated from inside to outside. This means that we need to evaluate innermost function and then the one outside it. This is an opposite order of evaluation with respect to domain which is evaluated from outside to inside. We shall highlight these aspects while working with examples.

In the following sections, we discuss various context of least and greatest values.

Standard functions

We are familiar with the least value, greatest value and range of the most standard functions of all origin. Consider constant, identity, reciprocal, modulus, greatest integer, least integer, fraction part, trigonometric, inverse trigonometric, exponential and logarithmic functions. All these functions have been described in detail and we know their properties with respect to least and greatest values and also the range. Greatest value of sine function, for example, is 1. On the other hand, exponential and logarithmic functions etc. neither have minimum (therefore least value) nor maximum (therefore greatest value). However, these functions have least and greatest in finite interval in accordance with mean value theorem.

In case, the function can be reduced to the standard forms having least and greatest values, then it is possible to know its range. In the example, we consider one such trigonometric function.

Problem : Find the range of the continuous function given by :

f x = a cos x + b sin x

where “a” and “b” are constants.

Solution : Here, given function is addition of two trigonometric functions. As we know least and greatest values of sine and cosine functions, we shall attempt to reduce given function in terms of either sine or cosine function (note that the algorithm for reducing addition of sine and cosine functions as presented here is a standard algorithm. We should also note that this algorithm, as a matter of fact, is used in analyzing superposition principle of waves) :

Let a = r cos α and b = r sin α


r = a 2 + b 2

Substituting in the given function, we have :

f x = r cos α cos x + r sin α sin x = r cos x α = a 2 + b 2 cos x α

We know that minimum and maximum values of cosine function are "-1" and "1" respectively. Hence,

f x min = - a 2 + b 2

f x max = a 2 + b 2

Therefore, range of the given function is :

Range = [ - a 2 + b 2 , a 2 + b 2 ]

Questions & Answers

where we get a research paper on Nano chemistry....?
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Maira Reply
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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I think
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Brian Reply
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scanning tunneling microscope
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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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
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The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Stoney Reply
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
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Period of sin^6 3x+ cos^6 3x
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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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