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, \(x=\text{ }0\) is also a local maximum; however, since f(0) has the largest value on the entire domain, \(x=\text{ }0\) 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:
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(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, for \(f\left( x \right) < f\left( a \right)\) 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. |
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(B) |
GLOBAL MAXIMUM: |
A point \(x = a\) is a global maximum for f(x) if \(f\left( x \right) \ge f\left( a \right)\) for all \(x \in D\) (the domain of f(x)). |
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(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 \(x=a,f\left( a \right)\) should be the minimum in that interval. |
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(D) |
GLOBAL MINIMUM: |
A point \(x = a\) is a global minimum for f(x) if \(f\left( x \right) \ge f\left( a \right)\) 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 \(x=\text{}0,f\left( x \right)\text{}=\text{}sinx\) has local (and global) maxima at \(\begin{align}x = 2n\pi + \frac{\pi }{2},n \in \mathbb{Z}\end{align}\) and local (and also global) minima at \(\begin{align}x = 2n\pi - \frac{\pi }{2},n \in \mathbb{Z}\end{align}\). Note that, for a function \(f\left( x \right),\) a local minimum could actually be larger than a local maximum elsewhere. There is no restriction to this. A local minimum value impliese 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.