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Above argument is valid for all continuous function which may have varying combination of increasing and decreasing trends within the domain of function. The function values at end points of a closed interval are extremums (minimum or maximum) - may not be least or greatest. In the general case, there may be more minimum and maximum values apart from the ones at the ends of closed interval. This generalization, as a matter of fact, is the basis of “extreme value theorem”.
The extreme value theorem of continuous function guarantees existence of minimum and maximum values in a closed interval. Mathematically, if f(x) is a continuous function in the closed interval [a,b], then there exists f(l) ≤ f(x) and f(g) ≥ f(x) such that f(l) is global minimum and f(g) is global maximum of function.
As discussed earlier, there at least exists a pair of minimum and maximum at the end points. There may be more extremums depending on the nature of graph in the interval.
If a function is continuous, then least i.e. global minimum, “A” and greatest i.e. global maximum, “B”, in the domain of function correspond to the end values specifying the range of function. The range of the function is :
$$[A,B]$$
If function is not continuous or if function can not assume certain values, then we need to suitably analyze function and modify the range given above. We shall discuss application of the concept of least and greatest values to determine range of function in a separate module.
There are three cases for determining minimum and maximum values. However, we should clearly underline that these methods give us relative minimum and relative maximum values – which may or may not be the greatest (global) or least (global) values. We need to interpret minimum and maximum in the context of specified domain to ascertain whether minimum and maximum are least and greatest respectively or not?
(i) function is differentiable in the domain of function.
(ii) function is continuous in the domain of function
(iii) function is discontinuous at certain points in the domain of function.
The derivative of function exists for all values of x in the domain. In this case, we follow the algorithm given here (without proof- its proof is based on Taylor’s expansion) :
1: Determine first derivative.
2: Equate derivative to zero.
3: Solve equation obtained in the step 2 for x.
4: If there is no real solution of equation, then function has no minimum or maximum.
5: If there is real solution of equation, then determine second derivative. Put root values in the expression of second derivative one after another and see whether second derivative is non-zero. If second order derivative is positive non-zero, then function is minimum at that root value. On the other hand, if second order derivative is negative non-zero, then function is maximum at that root value. We should note that these conclusions are valid for all higher even derivatives, which we might need to evaluate.
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