tgt

## 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 = 0$$f(x) = {x^2}$ has a local (and global) minimum at $x = 0$$f(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.