5.7 Least and greatest function values

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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 :

$\text{Range}=\left[\text{least, greatest}\right]=\left[\mathrm{a,b}\right]$

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 :

$f\left(x\right)=a\mathrm{cos}x+b\mathrm{sin}x$

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) :

$\text{Let}\phantom{\rule{1em}{0ex}}a=r\mathrm{cos}\alpha \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}b=r\mathrm{sin}\alpha$

where,

$r=\sqrt{\left({a}^{2}+{b}^{2}\right)}$

Substituting in the given function, we have :

$f\left(x\right)=r\mathrm{cos}\alpha \mathrm{cos}x+r\mathrm{sin}\alpha \mathrm{sin}x=r\mathrm{cos}\left(x-\alpha \right)=\sqrt{\left({a}^{2}+{b}^{2}\right)}\mathrm{cos}\left(x-\alpha \right)$

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

$f{\left(x\right)}_{\mathrm{min}}=-\sqrt{\left({a}^{2}+{b}^{2}\right)}$

$f{\left(x\right)}_{\mathrm{max}}=\sqrt{\left({a}^{2}+{b}^{2}\right)}$

Therefore, range of the given function is :

$\text{Range}=\left[-\sqrt{\left({a}^{2}+{b}^{2}\right)},\sqrt{\left({a}^{2}+{b}^{2}\right)}\right]$

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