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Sunday, 3 August 2014

CHAPTER 10 - Introduction to Monotonicity

Consider a function represented by the following graph:
For two different input arguments {x_1} and {x_2}, where {x_1} < {x_2},{y_1} = f\left( {{x_1}} \right) will always be less than {y_2} = f\left( {{x_2}} \right).
That is,
{x_1} < {x_2} implies f\left( {{x_1}} \right) < f\left( {{x_2}} \right)
Such a function is called a strictly increasing function or a monotonically increasing function (The word ‘monotonically’ apparently has its origin in the word monotonous; for example, a monotonous routine is one in which one follows the same routine repeatedly or continuously; similarly a monotonically increasing function is one that increases continuously).
Now, consider f\left( x \right) = \left[ x \right]. For this function
{x_1} < {x_2} does not always imply f\left( {{x_1}} \right) < f\left( {{x_2}} \right)
However,
{x_1} < {x_2} does imply f\left( {{x_1}} \right) \le f\left( {{x_2}} \right)
In other words, f\left( x \right) = \left[ x \right] is not strictly (or monotonically) increasing. It will nevertheless be termed increasing.
Now consider a function represented by the following graph:
For two different input arguments {x_1} and {x_2}, where {x_1} < {x_2},{y_1} = f\left( {{x_1}} \right) will always be greater than {y_2} = f\left( {{x_2}} \right).
That is,
{x_1} < {x_2} \Rightarrow f\left( {{x_1}} \right) > f\left( {{x_2}} \right)
Such a function is called a strictly decreasing function or a monotonically decreasing function.
Now consider f\left( x \right) =  - \left[ x \right]. For this function
{x_1} < {x_2} does not imply f\left( {{x_1}} \right) > f\left( {{x_2}} \right)
However,
{x_1} < {x_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f\left( {{x_1}} \right) \ge f\left( {{x_2}} \right)
Therefore, f\left( x \right) =  - \left[ x \right] is not strictly decreasing. It would only be termed decreasing.
The following table lists down a few examples of functions and their behaviour in different intervals. You are urged to verify all the assertions listed on your own.
Function
Behaviour
f\left( x \right) = x
Strictly increasing on \mathbb{R}
f\left( x \right) = {x^2}
Strictly decreasing on \left( { - \infty ,0} \right]
Strictly increasing on \left[ {0,\infty } \right)
f\left( x \right) = \sqrt x
Strictly increasing on \left[ {0,\infty } \right)
f\left( x \right) = {x^3}
Strictly increasing on \mathbb{R}
f\left( x \right) = \left| x \right|
Strictly decreasing on \left( { - \infty ,0} \right]
Strictly increasing on \left[ {0,\infty } \right)
f\left( x \right) = \dfrac{1}{x}
Neither increasing nor decreasing on \mathbb{R}.
Strictly decreasing on \left( { - \infty ,0} \right)
Strictly decreasing on \left( {0,\infty } \right)
f\left( x \right) = \left[ x \right]
Increasing on \mathbb{R}
f\left( x \right) = \left\{ x \right\}
Neither increasing nor decreasing on \mathbb{R}
However, strictly increasing on \left[ {n,n + 1} \right) where n \in\mathbb{Z}
f\left( x \right) = \sin x
Neither increasing nor decreasing on \mathbb{R}.
Strictly increasing on [(\,2n - \dfrac{1}{2}\,)\pi ,(2n + \dfrac{1}{2})\pi ];n \in\mathbb{Z}
Strictly decreasing on [(2n + \dfrac{1}{2})\pi ,(2n + \dfrac{3}{2})\pi ];n \in\mathbb{Z}
f\left( x \right) = \cos x
Neither increasing nor decreasing on \mathbb{R}.
Strictly increasing on [(2n - 1)\pi ,2n\pi ];n \in\mathbb{Z}
Strictly decreasing on [2n\pi ,(2n + 1)\pi ];n \in\mathbb{Z}
f\left( x \right) = \tan x
Neither increasing nor decreasing on \mathbb{R} .Strictly increasing
on \left( {\left( {n - \dfrac{1}{2}} \right)\pi ,\left( {n + \dfrac{1}{2}} \right)\pi } \right);n \in\mathbb{Z}
f\left( x \right) = {e^x}
Strictly increasing on \mathbb{R}
f\left( x \right) = {e^{ - x}}
Strictly decreasing on \mathbb{R}
f\left( x \right) = \ln x
Strictly increasing on \left( {0,\infty } \right)
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