Although the name itself is suggestive, we introduce the concept of maxima and minima here through a simple example:
Consider an arbitrary function .
The concept of Maxima and Minima is a way to characterize the peaks and troughs of . For example, we see that there is a peak at ; this point is therefore a local maximum; similarly, is also a local maximum; however, since has the largest value on the entire domain, is also a global maximum.
Analogously, and are local minimum points; is also a global minimum.
Having introduced the concept intuitively, we can now introduce more rigorous definitions:
(A) LOCAL MAXIMUM:
A point is a local maximum for if in the neighbourhood of i.e in where can be made arbitrarily small, for all . This simply means that if we consider a small region (interval) around should be the maximum in that interval.
(B) GLOBAL MAXIMUM:
A point is a global maximum for if for all (the domain of ).
(C) LOCAL MINIMUM:
A point is a local minimum for if in the neighbourhood of , i.e. in , (where can have arbitrarily small values), for all . This means that if we consider a small interval around should be the minimum in that interval
(D) GLOBAL MINIMUM:
A point is a global minimum for if for all (the domain of ).
As examples, has a local (and global) minimum at , has a local (and global) minimum at , has local (and global) maxima at and local (and also global) minima at . Note that, for a function , a local minimum could actually be larger than a local maximum elsewhere. There is no restriction to this. A local minimum value implies a minimum only in the immediate ‘surroundings’ or ‘neighbourhood’ and not ‘globally’; similar is the case for a local maximum point.
To proceed further, we now restrict our attention only to continuous and differentiable functions.