5.6 Minimum and maximum values  (Page 3/6)

$$f\left(m\right)<f\left(m+h\right)\text{and}f\left(m\right)<f\left(m-h\right)\text{as}h\to 0$$

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,

$$f\left(m\right)>f\left(m+h\right)\text{and}f\left(m\right)>f\left(m-h\right)\text{as}h\to 0$$

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,

$$f\left(l\right)\le f\left(x\right)\text{for all}x\in [a,b]$$

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,

$$f\left(l\right)\le f\left(x\right)\text{for all}x\in (a,b)$$

$$f\left(l\right)\le \underset{x\to a+0}{\overset{}{\lim }}f\left(x\right)$$

$$f\left(l\right)\le \underset{x\to b-0}{\overset{}{\lim }}f\left(x\right)$$

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,

$$f\left(g\right)\ge f\left(x\right)\text{for all}x\in [a,b]$$

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,

$$f\left(g\right)\ge f\left(x\right)\text{for all}x\in (a,b)$$

$$f\left(g\right)\ge \underset{x\to a+0}{\overset{}{\lim }}f\left(x\right)$$

$$f\left(g\right)\ge \underset{x\to b-0}{\overset{}{\lim }}f\left(x\right)$$

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 ${\log }_{e}x$ . It extends from negative infinity to positive infinity, if base is greater than 1. The function is a strictly increasing function in its entire domain. As such, it has not a single minimum or maximum. The extremely large values at the domain ends can not be considered to be extremum as we can always have function values greater or less than one considered to be maximum or minimum. This argument is valid for behavior of functions near end points of an open interval domain. There can always be values greater or smaller than one considered.

However, nature of graph with respect to extremum immediately changes when we define same logarithmic function in a closed interval say [3,4], then ${\log }_{e}3$ and $\log {}_{e}4$ are the respective local minimum and maximum. Incidentally since function is strictly increasing in the domain and hence in the sub-interval, these extremums are global i.e. end values of function are global minimum and maximum in the new domain of the function.