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 |
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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.

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

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

Graph of function |
---|

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.

** Critical points **

Critical points are those points where minimum or maximum of a function can occur. We see that minimum and maximum of function can occur at following points :

(a) Points on the graph of function, where derivative of function is zero. At these points, function is continuous, limit of function exists and tangent to the curve is parallel to x-axis.

(b) Points where function is continuous but not differentiable. Limit of function exits at those points and are equal to function values. Consider, for example, the corner of modulus function graph at x=0. Minimum of function exist at the corner point i.e at x=0.

(c) Points where function is discontinuous (note that discontinuous is not undefined). A function has function value at the point where it is discontinuous. Neither limit nor derivative exists at discontinuities. Example : piece-wise defined functions like greatest integer function, fraction part function etc.

We can summarize that critical points are those points where (i) derivative of function does not exist or (ii) derivative of function is equal to zero. The first statement covers the cases described at (b) and (c) above. The second statement covers the case described at (a). We should, however, be careful to interpret definition of critical points. These are points where minimum and maximum “can” exist – not that they will exist. Consider the graph shown below, which has an inflexion point at “A”. The tangent crosses through the graph at inflexion point. In the illustration, tangent is also parallel to x-axis. The derivative of function, therefore, is zero. But “A” is neither a minimum nor a maximum.

Graph of function |
---|

Thus, minimum or maximum of function occur necessarily at critical points, but not all critical points correspond to minimum or maximum.

Note : We need to underline that concept of critical points as explained above is different to the concept of critical points used in drawing sign scheme/ diagram.

**Graphical view**

There are mathematical frameworks to describe and understand nature of function with respect to minimum and maximum. We can, however, consider a graphical but effective description that may help us understand occurrence of minimum and maximum values. We need to understand one simple fact that we can have graphs of any nature except for two situations :

1: function is not defined at certain points or in sub-intervals.

2: function can not be one-many relation. In this case, the given relation is not a function in the first place.

Clearly, there exists possibility of minimum and maximum at all points on the continuous portion of function where derivative is zero and at points where curve is discontinuous. This gives us a pictorially way to visualize where minimum and maximum can occur. The figure, here, shows one such maximum value at dicontinuity.

Graph of function |
---|

** Relative or local minimum and maximum **

The idea of local or relative minimum and maximum is clearly understood from graphical representation. The minimum function value at a point is least in the immediate neighborhood where minimum occurs. A function has a relative minimum at a point x=m, if function values in the immediate neighborhood on either side of point are less than the value at the point. To be precise, the immediate neighborhood needs to be infinitesimally close. Mathematically,

The maximum function value at a point is greatest in the immediate neighborhood where maximum occurs. A function has a relative maximum at a point x=m, if function values in the immediate neighborhood on either side of point are greater than the value at the point. To be precise, the immediate neighborhood needs to be infinitesimally close. Mathematically,

** Global minimum and maximum **

Global minimum is also known by “least value” or “absolute minimum”. A function has one global minimum in the domain [a,b]. Global minimum, f(l), is either less than or equal to all function values in the domain. Thus,

If the domain interval is open like (a,b), then global minimum, f(l), also needs to be less than or equal to function value, which is infinitesimally close to boundary values. It is because open interval by virtue of its inequality does not ensure this. What we mean that it does not indicate how close “x” is to the boundary values. Hence,

Similarly, global maximum is also known by “greatest value” and “absolute maximum”. A function has one global maximum in the domain [a,b]. Global maximum, f(g), is either greater than or equal to all function values in the domain. Thus,

If the domain interval is open like (a,b), then global maximum, f(m), also needs to be greater than or equal to function value, which is infinitesimally close to boundary values. It is because open interval by virtue of its inequality does not ensure this. Hence,

** Domain interval **

Nature of domain interval plays an important role in deciding about occurrence of minimum and maximum and their nature. In order to understand this, we need to first understand that the notion of very large positive value and concept of maximum are two different concepts. Similarly, the notion of very large negative value and concept of minimum are two different concepts. The main difference is that very large negative or positive values are not finite but extremums are finite. Consider a natural logarithmic graph of

Definite sub-interval of logarithmic function |
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However, nature of graph with respect to extremum immediately changes when we define same logarithmic function in a closed interval say [3,4], then

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”.

** 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.

** Range of function **

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 :

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.