CHAPTER 13- Introduction to Maxima and Minima

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.