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Sunday, 3 August 2014

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 f(x).
The concept of Maxima and Minima is a way to characterize the peaks and troughs of f(x). For example, we see that there is a peak at x = a; this point is therefore a local maximum; similarly, x = 0 is also a local maximum; however, since f(0) has the largest value on the entire domain, x = 0 is also a global maximum.
Analogously, x = b and x = c are local minimum points; x = c is also a global minimum.
Having introduced the concept intuitively, we can now introduce more rigorous definitions:
(A) \,\, LOCAL MAXIMUM:
A point x = a is a local maximum for f(x) if in the neighbourhood of a i.e in \left( {a - \delta ,a + \delta } \right) where \delta  can be made arbitrarily small, f\left( x \right) < f\left( a \right) for all x \in \left( {a - \delta ,a + \delta } \right)\backslash \left\{ a \right\}. This simply means that if we consider a small region (interval) around x = a,f(a) should be the maximum in that interval.
(B) \,\, GLOBAL MAXIMUM:
A point x = a is a global maximum for f(x) if f(x) \le f(a) for all x \in D (the domain of f(x)).
(C) \,\, LOCAL MINIMUM:
A point x = a is a local minimum for f(x) if in the neighbourhood of a, i.e. in \left( {a - \delta ,a + \delta } \right), (where \delta  can have arbitrarily small values), f\left( x \right) > f\left( a \right) for all x \in \left( {a - \delta ,a + \delta } \right)\backslash \left\{ a \right\}. This means that if we consider a small interval around x = a,f(a) should be the minimum in that interval
(D) \,\, GLOBAL MINIMUM:
A point x = a is a global minimum for f(x) if f(x) \ge f(a) for all x \in D (the domain of f(x)).
As examples, f\left( x \right) = \left| x \right| has a local (and global) minimum at x = 0f(x) = {x^2} has a local (and global) minimum at x = 0f(x) = \sin x has local (and global) maxima at x = 2n\pi  + \dfrac{\pi }{2},n \in\mathbb{Z} and local (and also global) minima at x = 2n\pi  - \dfrac{\pi }{2},n \in\mathbb{Z}. Note that, for a function f(x), 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.
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